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Technische Universitat Munchen

Ludwig-Maximilians-Universitat Munchen

On the structure of positivemaps

A thesis completed for the degree ofMaster of Science

in the Elite-Master’s program TMP by

Martin Idelfrom Krefeld

September 30, 2013

Supervisor: Prof. Dr. Michael M. Wolf

Date of Defence: October 15, 2013First Reviewer: Prof. Dr. Michael M. WolfSecond Reviewer: Prof. Dr. Michael Keyl

Hiermit erklare ich, dass ich die vorliegende Masterarbeit selbstandig angefertigt habe und nurdie angegebenen Quellen verwendet habe.

Garching, den 09.10.2013

Abstract

This thesis discusses two unrelated questions concerning positive linear maps between matrixalgebras. Positive maps are found in quantum information and provide the tool to determinewhether a state is seperable or not. In the first part of the thesis, we derive sufficient conditions fora normal form for certain classes of positive maps and provide hints as to what could be necessaryand sufficient conditions. The second and main part of the thesis is then concerned with fixedpoints of (unital or trace-preserving) positive maps, thereby generalizing known results for thefixed point spaces of trace-preserving completely positive maps, which form the basic conceptin quantum information theory (also called quantum channels). We first review basic facts andprove that the fixed point spaces of unital positive maps are Jordan algebras, before we provide adetailed study of Jordan subalgebras in finite dimensions (based on and extending the study byJacobson and Jacobson) and construct projections onto Jordan algebras (largely based on a seriesof papers by Størmer and extending and simplifying them to finite dimensions), which gives usa classification of the fixed points of trace-preserving positive maps, as well as certain resultsconcerning their peripheral spectrum.

Acknowledgements

I thank my advisor Prof. Wolf for giving me the opportunity to do my Master’s thesis in his groupdespite my (prior to starting the thesis) complete ignorance of his main areas of research and Ithank him for providing me with a topic.I am very much indebted to Alexander Muller-Hermes for many discussions. He helped me tofind literature on the topic and proofread most of my work. In addition, he helped me find someshortcuts at certain proofs (especially for the chapter on representation theory, where he made meaware of the fact I proved some things twice) and was very supportive and confident that I wouldget some nice results even when I wasn’t.Moreover, I am grateful to my friend Andres Goens-Jokisch, who helped me figure out somedetails in the proof of lemma 3.22.Another constant support came from my friend Javier Cuesta, who shared an office with mefor most of my thesis, and whom I would like to thank for many discussions, the proofreadingof lemma 3.27 some night in August, when I couldn’t see my mistakes anymore and countlessdiscussions. I also have to thank him for being one of the people to get me interested in quantuminformation in the first place.I owe special thanks to Robert Idel and Tobias Ried for reading parts of the first draft andcommenting on mistakes and the writing.Likewise, I would like to thank Sefano Duca, Emmanuel Klinger and Patrik Omland for listeningto my moaning, especially about the proof of 4.5 (which they tried to help me with).Last but not least, I would like to thank family and other friends for their support during the pastyear.

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Contents

Introduction 1

1. A Normal Form for Positive Maps 31.1. Classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Linear maps between matrix spaces . . . . . . . . . . . . . . . . . . . . . . . 4

2. Fixed Point Spaces of positive, unital maps 112.1. Basic notions of Jordan algebras . . . . . . . . . . . . . . . . . . . . . . . . . 142.2. Fixed point spaces of positive maps . . . . . . . . . . . . . . . . . . . . . . . 18

3. Representation theory of Jordan algebras 233.1. Associative specializations and embeddings of formally real algebras . . . . . . 263.2. Reduction of the representation theory to universal algebras . . . . . . . . . . . 293.3. Universal embeddings of finite dimensional Jordan algebras . . . . . . . . . . 333.4. Representations of the different types of Jordan algebras . . . . . . . . . . . . 45

4. Positive Projections onto fixed-point algebras 554.1. Existence and uniqueness of projections onto Jordan algebras . . . . . . . . . . 574.2. Projections onto direct sums with multiplicity . . . . . . . . . . . . . . . . . . 584.3. Projections onto spin factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4. Construction of positive projections onto spin factors . . . . . . . . . . . . . . 704.5. Projections onto reversible Jordan algebras . . . . . . . . . . . . . . . . . . . . 804.6. Construction of positive projections onto reversible Jordan algebras . . . . . . 88

5. Characterizations of positive projections onto fixed point algebras 945.1. Decomposability of unital projections onto Jordan algebras . . . . . . . . . . . 955.2. Properties of indecomposable projections onto spin factors . . . . . . . . . . . 97

6. Fixed point spaces of positive, trace-preserving maps and the peripheral spec-trum 1006.1. Fixed point spaces of positive, trace-preserving maps . . . . . . . . . . . . . . 1006.2. The peripheral spectrum of positive, trace-preserving maps . . . . . . . . . . . 103

Conclusion 110

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Martin Idel Contents

Bibliography 112

A. Appendix 115A.1. Spectral and Minimal Projections in Jordan Algebras . . . . . . . . . . . . . . 115A.2. Omitted proofs from chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 121

II

Introduction

The goal of this thesis is a better understanding of the structure of positive maps, which playa role in quantum information. The title of this thesis ”On the structure of positive maps“ isextremely broad, which is due to the fact that it covers two very different topics, which are onlyconnected by the fact that both of them concern positive maps.The first topic, which could be seen as the ”Theoretikum“, is treated in chapter 1 and generalizesa normal form for matrices (stated in [31]) to positive maps. Normal forms are interestingmathematical objects, because they give an insight to the structure of the objects considered andmay help proving other results. However, we will not consider any applications here.The rest of the thesis is then concerned with the structure of fixed point spaces of positive mapsand the projections onto fixed point spaces. The problem is well studied for completely positivemaps (see e.g. [40]) and recently, there has also been some work from a different angle in [29].Since this thesis is completed in the study program ”Theoretical and Mathematical Physics“,although it is mostly purely mathematical, we will review some physical motivations to study thisproblem.In principle, quantum mechanics could be seen as the study of completely positive maps, sincethe time evolution of the Schrodinger equation results in such maps and physical reasoning ongeneral maps leads to the notion of completely positive maps (see [26], chapter 8). Also, the studyof completely positive maps is much simpler than the study of positive maps, which is why muchmore literature is devoted to the subject. However, one of the most prominent tasks in the earlydevelopment of quantum information theory in the 1990s was to decide whether a given stateis separable or not (in the sense that its density matrix, which is a positive semidefinite matrixρ ∈ B(H)⊗B(H ′) for some Hilbert spacesH ,H ′, can be written as a sum of tensor products). In[14] it was proven that for every nonseparable state, there is a so-called ”entanglement witness“,which is a positive matrix W ∈ B(H)⊗B(H ′), such that tr(ρW) < 0. This W then corresponds toa positive, but not completely positive map. This resulted in a renewed interest in positive mapswith the goal to find structure theorems for the set of positive maps, in order to compute for agiven state, whether it is separable or not. One positive but not completely positive map is thetransposition, which gives rise to the Peres criterion (also called PPT-criterion, [27]), which iseasy to compute. It was also shown by the Horodeckis in [14] that every witness resulting fromdecomposable maps (i.e. maps that are a sum of completely positive and completely copositivemaps) is not stronger than the Peres criterion. In consequence, there was an increased interest infinding and classifying indecomposable maps. However, the program of complete classificationhas come to an end, since Gurwitz showed in [9] that the decision problem is supposedlyintractable (in the sense that it is NP-hard, which is believed to mean that a it cannot be computed

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Martin Idel Contents

efficiently on a computer). This means, assuming that the class NP is really intractable that astructure theorem for positive maps that gives an efficient way to prove separability is impossible.Since then, the interest in positive maps, at least from the physics community, seems to havedeclined somewhat, although many open problems remain.While for completely positive maps, it turns out that the fixed point space is roughly a C*-algebra(at least in finite dimensions) and fixed point projections are therefore given by projections ontoC*-algebras, this is no longer true for arbitrary positive maps. It turns out that the algebraicnotion needed here is a so-called Jordan algebra.The main part of the thesis does not assume any prior knowledge on Jordan algebras, while it doesassume some prior knowledge of C*-algebras. With this background, the theory is developedand almost all of the proofs needed are given along the way, with the exception of one technicalproposition on Jordan algebras (proposition 2.12), lemma 2.6 for the peripheral spectrum, whichhas a technical proof using approximations to diophantine equations, which we do not want torecite here, and some main structure theorems for Jordan algebras (theorem 2.13) and associativealgebras (Artin-Wedderburn theorem 3.23, and Skolem-Noether theorem 3.15), which have verylong, but well-known proofs. Other than this, sources for the proofs and propositions are givenand the proofs are included here. Parts of chapter 4 need a lot of knowledge about projections inJordan algebras. Since this knowledge is not central for the rest of the thesis, we have included itin a different section in appendix A.The structure of the main part is as follows: After reviewing a lot of basic material on fixed points,Jordan algebras and positive maps in chapter 2, we classify all Jordan algebras that are subalgebrasof complex matrices in chapter 3, while constructing and characterizing all projections onto theseJordan algebras in chapters 4 and 5. Finally, this gives us a tool to further elaborate on fixed pointspaces of positive maps. In order to better guide the reader, from chapter 3 onwards, we haveassembled the main results of the chapter in the beginning and outlined the ideas of their proofs.

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1. A Normal Form for Positive Maps

The idea of this part of the thesis is to generalize a normal form for certain classes of nonnegativematrices to positive maps on matrices. The resulting normal form (see theorem 1.15) is knownfor completely positive maps ([40]) and we will generalize this to some extent, yet necessary andsufficient conditions are only conjectured. Some proofs are omitted, because they are deemedeasy or known, but for the reader’s convenience, we have gathered most of them in appendix A.2.

1.1. Classical results

Let us first introduce some notation to state the classical result we wish to generalize:

Definition 1.1. A positive semidefinite matrix A ∈ Md (d × d complex matrices) is called fullyindecomposable, if there do not exist permutation matrices P, Q and square matrices A1, A2without zero-rows or columns, such that

QAP =

(A1 0B A2

)Basically this implies that we have an invariant subspace for the matrix after independentpermutation of rows and columns. This is a stronger notion than the notion of irreducibility of amatrix, where P = Q.

With this definition, we state the result, which was in a less general form first introduced in [31],while we use the result with necessary and sufficient conditions similar to [4]:

Theorem 1.2. For a nonnegative matrix A : Rn → Rn there exist diagonal matrices D1,D2 suchthat D1AD2 is doubly stochastic iff there exist permutation matrices P and Q such that PAQ is adirect sum of fully indecomposable matrices.

There have been several proofs of this fact in the literature with different methods. We willgeneralize certain ideas of the proof of [4], which builds on material from [23]. Other proofs aregiven in [20], [21] and [32].However, since we cannot yet give a full generalization with necessary and sufficient conditions,we will collect the different equivalent versions of this theorem given in the literature:

Proposition 1.3. In the setting of theorem 1.2, we have the following equivalence:

• A is fully indecomposable

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Martin Idel Linear maps between matrix spaces

• [32]: A has total support, i.e. every positive element lies on a positive diagonal, wherea diagonal is given by a permutation σ and the list of elements A1σ(1), . . . Adσ(d). Thediagonal is positive, if all entries are positive.

• [20]: A has a doubly stochastic pattern, i.e. there exists a doubly stochastic matrix B suchthat B has nonzero entries exactly where A has nonzero entries.

• [4]: A is irreducible with positive main diagonal.

The proofs of equivalence of a cited notion to the first statement can be found in the cited papers.

1.2. Linear maps between matrix spaces

Definition 1.4. Fix the following notation:

• Md d2-dimensional complex square matrices

• Cd cone of d2-dim. positive definite matrices

• Cd cone of d2-dim. positive semidefinite matrices

The last definition is related to the fact that the semidefinite matrices are the closure of the positivedefinite matrices. We write A ≥ 0 for A ∈ Cd.

First recall the basic definition of positive and doubly stochastic maps:

Definition 1.5. A linear map E :Md →Md is called positive, if for all A ≥ 0, A ∈ Md we haveE(A) ≥ 0.A positive linear map E as above is called doubly stochastic, if E(1) ∝ 1 and E∗(1) ∝ 1 , whereE∗ denotes the adjoint with respect to the usual inner product defined through the matrix trace.

Then we define strictly kernel-reducing maps:

Definition 1.6. Let E : Md → Md be a positive, linear map. Then E is called strictly kernel-reducing if for all singular A ≥ 0, dim kerE(A) < dim ker A.

In addition, we define fully indecomposable maps. In analogy to the classical definition, we say:

Definition 1.7. A positive map is called fully indecomposable if

E(PMdP) ⊂ QMdQ

for Q and P projections with the same dimension implies that P,Q ∈ 0,1.

We remark that when P = Q, this is exactly the notion of irreducibility of matrices as definedin [40] (chapter 6). For the reader’s convenience, we recall this definition and some basic facts,which we do not prove here:

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Proposition 1.8 ([40], theorem 6.2). For a positive, linear map T : Md → Md the followingproperties are equivalent:

• T is irreducible, i.e. if P ∈ Md is a Hermitian projector such that T (PMdP) ⊂ PMdPthen P ∈ 0,1

• For every nonzero A ≥ 0 we have (id + T )d−1(A) > 0.

• For every nonzero A ≥ 0 and every strictly positive t ∈ R we have exp(tT )(A) > 0

We also recall a version of Brouwer’s fixed point theorem, which will be used later:

Proposition 1.9 (Brouwer’s fixed point theorem). Every continuous map T : C → C, where C isa convex, compact space, has a fixed point, i.e. a point x ∈ C such that T (x) = x.

In order to obtain a normal form in the above spirit, we generalize the Menon-Operator introducedin [23] to positive maps:

Definition 1.10. Let E : Md → Md be a positive map, such that E,E∗ : Cd → Cd. LetD : Cd → Cd denote matrix inversion, then we define the following operator:

G : Cd → Cd

G(·) :=D E∗ D E(·)

tr(D E∗ D E(·))

This operator is well-defined and sends positive definite matrices of trace one onto itself.

First we remark that the conditions E,E∗ : Cd → Cd are necessary for a generalized normal formto exist:

Proposition 1.11. Let E : Md → Md be a positive linear map. The following statements areequivalent:

(1) E(1) > 0

(2) E : Cd → Cd (equivalence to the first item stated without proof in [1]).

(3) for X,Y > 0, YE(X(·)X†)Y† : Cd → Cd.

(4) E∗(X) , 0 for all X ≥ 0

Proof. See proposition A.13.

Lemma 1.12. There exist invertible X,Y ∈ Md such that Y−1E(X(·)X†)Y−† is a doubly stochas-tic map iff G has a fixed point (eigenvalue) in the set of positive definite trace one matrices.Furthermore, X,Y can be chosen such that X,Y > 0.

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Proof. The last proposition 1.11 confirms in particular that E,E∗ : Cd → Cd are necessaryconditions for such a normal form to exist. Hence, we restrict to maps, where E,E∗ : Cd → Cd,whence the Menon operator is well-defined.Let ρ > 0 be the fixed point of G. Then define 0 < σ := E(ρ). Since ρ is a fixed point, oneimmediately sees that E∗(σ−1) = ρ−1. Now define X :=

√ρ and Y :=

√σ (i.e. XX† = ρ and

YY† = σ), then X,Y are positive definite and if we define the map:

E′ :Md →Md

E′(·) := Y−1E(X(·)X†)Y−†

then a quick calculation shows E′(1) = 1 and E′∗(1) = 1:

E′(1) = Y−1E(X(·)X†)Y−† = Y−1E(ρ)Y−†

= Y−1σY−† = Y−1YY†Y−† = 1

(likewise for E′∗(1)).Conversely, given X,Y as in the lemma, XX† would be a fixed point of the Menon-operator.

We observe:

Lemma 1.13. The Menon-Operator defined above is a continuous, homogeneous positive map

Proof. Obviously, via the linearity of E and E∗ and since any scalar multiple is inverted by matrixinversion, the Menon-Operator is homogeneous, since we have two matrix inversions. It is alsoorder-preserving (i.e. positive), since matrix inversion is order reversing and E and its adjoint areorder preserving. The map is continuous, since matrix inversion is continuous and linear maps onfinite dimensional spaces are also continuous.

Now we can give sufficient conditions for the existence of such a normal form:

Lemma 1.14. Let E :Md →Md be a positive map such that E : Cd → Cd and E∗(1) > 0, thenthere exist maps X,Y > 0 such that Y−1E(X(·)X†)Y−† is a doubly stochastic map.

Proof. The condition that E : Cd → Cd makes G a well defined map on Cd → Cd, which sendsthe set of semidefinite unit trace matrices into itself. Since this is a compact set, by Brouwer’sfixed point theorem, the map G has a fixed point having necessarily full rank as G maps tofull-rank matrices.

For the classical case, one idea (c.f. [4]) is to extend the map G to Cd for all cases, however, thisis not as easy in our case, since Cd is not a polyhedral cone (see e.g. [19], for polyhedral cones, acontinuous extension to the closure always exist, however this is not the case for other cones).Hence the ideas of [4], leading to sufficient and necessary conditions of the classical case cannotbe used here.We can also give a stronger set of sufficient conditions:

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Theorem 1.15. Given a positive trace-preserving map E such that there exists an ε > 0 such thatfor all matrices ρ ≥ ε1 with unit trace it holds that E(ρ) ≥ ε

1+(d−1)ε1, then we can find X,Y > 0such that Y−1E(X(·)X†)Y−† is a doubly stochastic map. Furthermore, this result comprises thelast lemma.

Proof. First observe that for trace-preserving E, the adjoint map is unital. Now assume E(ρ) ≥ δ1,then:

ρ ≥ ε1 ⇒ E(ρ) ≥ δ1 ⇒ D(E(ρ)) ≤1δ1

⇒ E∗(D(E(ρ))) ≤1δ1 ⇒ D(E∗(D(E(ρ)))) ≥ δ1

where we used the unitality of E∗ in the third step. Since we assumed tr(ρ) = 1, we have thatthe maximal eigenvalue λmax of E(ρ) fulfills λmax ≤ 1 − (d − 1)δ. Hence we have δ1/d ≤ ρ ≤(1 − (d − 1)δ)1/d and we obtain via the same reasoning:

tr(D(E∗(D(E(ρ))))) ≤ (1 − (d − 1)δ) tr(1)

This implies

G(ρ) ≥δ

1 − (d − 1)δ1/d

Now we want δ1−(d−1)δ ≥ εd, in which case the compact set of matrices ρ > 0| tr(ρ) = 1, ρ ≥ ε1/d

is mapped into itself, hence by Brower’s fixed point theorem, we obtain a positive definite fixedpoint of G. Therefore:

δ

1 − (d − 1)δ≥ εd

⇔dε

1 + (d − 1)εd≤ δ

Finally, we need to see that this is indeed an extension of the above result. To this end letE : C

d→ Cd, then for all density matrices ρ ≥ 0 there exists a δρ such that E(ρ) ≥ δρ1. But the

set of density matrices is compact, hence the minimum δ := mintr(ρ)=1 δρ is attained and thereforeδ > 0, thus for all ρ ≥ 0, tr(ρ) = 1 we have E(ρ) ≥ δ1. In consequence for ε small enough, suchthat δ ≥ εd

1+(d−1)εd , the implication of our theorem holds.

We would now like to find necessary conditions. This has proven unsuccessful so far, but we canprovide a number of maybe useful observations:

Lemma 1.16 (noted without proof in [40]). Given A ≥ 0 such that supp(E(A)) ⊂ supp(A), thenE(PMdP) ⊂ PMdP, where P is the orthogonal projector onto the support of A.

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Proof. See lemma A.14

Lemma 1.17 (noted without proof in [40]). E is irreducible iff there does not exist a nontrivialorthogonal projection P such that tr(E(P)(1 − P)) = 0.


Lemma 1.18. For a doubly stochastic map T , we have that T is a sum of irreducible maps.

Proof. Let P be a nontrivial Hermitian projector reducing T (if no such projector exists, we arefinished). Then, by the above lemma, tr(T (P)(1 − P)) = 0. But:

0 = tr(T (P)(1 − P)) = tr(PT ∗(1 − P))

= tr(P) − tr(PT ∗(P)) = tr(P) − tr(PT (P))

= tr(PT (1 − P))

where we used that T is doubly-stochastic in the second and last equality and in between weonly used the cyclicity and linearity of the trace. This means that if P reduces T , then also 1 − Preduces T , i.e.

T (PMdP) ⊂ PMdP

T (P⊥MdP⊥) ⊂ P⊥MdP⊥

which implies that T is a direct sum of maps defined on PMdP and P⊥MdP⊥.We obtain these maps by T1 := T (P.P)|PMdP and T2 := T (P⊥.P⊥)P⊥MdP⊥ . Obviously, P, P⊥

are the identities on the respective subspaces and the maps are therefore doubly stochastic, i.e.T1(1PMdP) = T (P) = P = 1PMdP (and for T2 equivalently).We can iterate the procedure, which will obviously terminate after finitely many steps, thus givinga reduction to a direct sum of doubly-stochastic irreducible maps.

This condition will not turn out to be necessary, although we have the following result:

Lemma 1.19 ([40]). Irreducibility is invariant under similarity transformations, i.e. for C > 0 Eis irreducible iff CE(C−1(·)C−†)C† is irreducible.


However, a map of the form YE(·)Y† is not necessarily irreducible if E is irreducible, whichimplies that irreducibility is not necessarily preserved under the equivalence relation we areconsidering. A stronger (and invariant) notion would be the one of being strictly kernel-reducing,which will play a role later on:

Lemma 1.20. Let E be a positive, linear map, then it is a strictly kernel-reducing map iff for anarbitrary invertible Y also YE(·)Y† is a kernel-reducing map.

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Proof. Since Y is invertible, it does not change the dimension of the kernel.

Proposition 1.21. Strictly kernel-reducing positive linear maps are always irreducible. Hence, ifE is a positive, linear map, then it is irreducible and kernel-reducing iff for any Y > 0 the mapYE(·)Y† is irreducible and kernel-reducing.

Proof. Let E be irreducible and strictly kernel-reducing, then YE(·)Y† is still strictly kernel-reducing by the above lemma and obviously ker(Id +YE(·)Y†)(A) ⊂ ker A for every A ≥ 0 wherethe inclusion is strict if A has not full rank. This implies that (Id +YE(·)Y†)d−1(A) > 0, which isequivalent to irreducibility.

In the following observation, we clarify why irreducibility is not invariant under our equivalencerelation:

Lemma 1.22. If E in the above setting is not strictly kernel-reducing, then there exists aninvertible Y such that YE(·)Y† is reducible, hence irreducibility is no invariant of this kind ofoperation.

Proof. Let E be irreducible. Now suppose E is not strictly kernel-reducing. This means thereexists A ≥ 0 singular matrix such that dim kerE(A) ≥ dim ker A. Since the kernels are vectorspaces, this implies we can find a unitary matrix U transforming the basis such that ker(E(A)) ⊃U · ker A. This implies:

ker(E(A)) ⊃ U · ker A

⇔ ker(E(A)) ⊃ ker UAU†

⇔ ker(U†E(A)U) ⊃ ker A

The latter implies supp(U†E(A)U) ⊂ supp A, which means that the map E′(·) := U†E(·)U isreducible by the projection P onto the support of A. This is a contradiction.

This can be used to obtain the following corollaries:

Corollary 1.23. If T is a doubly-stochastic irreducible positive linear map, then there exists aunitary matrix U such that UT (·)U† is a direct sum of strictly kernel-reducing or one-dimensionalmaps.

Proof. Note that for an arbitrary unitary U > 0 the maps T (U(·)U†) and UT (·)U† are stilldoubly-stochastic. Now suppose we can find U such that UT (·)U† is reducible. Since UT (·)U†

is still doubly stochastic as U is unitary, by Lemma 1.18, UT (·)U† is a direct sum of (possiblykernel non-reducing) doubly stochastic maps. We can now iterate the procedure, obtainingU1 ⊕ U2 with unitary U1,2 such that (U1 ⊕ U2)UT (·)U†(U1 ⊕ U2)† is a direct sum of (possiblykernel non-reducing) doubly stochastic maps. Also, (U1 ⊕ U2)U is unitary. Since we are in finitedimensions, this procedure terminates and we obtain a unitary U such that UT (·)U† is a directsum of irreducible and strictly kernel-reducing (or one-dimensional) maps.

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Corollary 1.24. A positive map E is kernel-reducing iff Y−1E(X(·)X†)Y−† is kernel-reducing forevery invertible, Hermitian X,Y.

Proof. Lemma 1.19 together with the fact that being strictly kernel-reducing is an invariantof similarity transformations, which can easily be obtained from lemma 1.20, implies that Eis strictly kernel-reducing iff C−1E(C(·)C†)C−† is (where C is Hermitian, invertible), which,via proposition 1.21 and again lemma 1.20, holds iff for an arbitrary invertible Hermitian ZZ−1C−1E(C(·)C†)C−†Z−†.Choosing C := X and Z := C−1Y , we obtain that E is strictly kernel-reducing iff Y−1E(X(·)X†)Y−†

is strictly kernel-reducing.

Along the way, we have also proven the following observation connecting our two definitions ofkernel-reducing and fully indecomposable maps:

Proposition 1.25. A positive map E is strictly kernel-reducing if and only if it is fully indecom-posable.

This leads us to the following conjecture, partial results of which have been proven above:

Conjecture 1.26. Consider a positive linear map E : Md → Md. Then there exist positivedefinite matrices X,Y ∈ Cd, such that E′ := Y−†E(X(·)X†)Y−1 is a doubly stochastic map iffE(1) > 0, E∗(1) > 0 and UE(·)U† is a direct sum of fully indecomposable maps for some unitaryU.

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2. Fixed Point Spaces of positive, unitalmaps

This chapter sets the stage for the main objective of the rest of the thesis: to obtain a characteriza-tion of the fixed points of certain classes of positive maps. Let us first fix some notation and thenintroduce the main objects of interest, in particular the set of fixed points of an arbitrary positivemap and its peripheral spectrum.

Notation:

Throughout this thesis,Md,d′ =Md,d′(C) shall be the set of d × d′ matrices with complex entries,where d, d′ ∈ N. For shorter notation, we will also introduce the notationMd :=Md,d for squarematrices (our main interest). Matrices over other (skew-)fields will always be written with theirfield, e.g. Md(R) andMd(H). The set (Md)h will then denote the Hermitian matrices ofMd andin general, the subscript h will indicate that we take the self-adjoint elements of the respective set(in which sense the matrices are self-adjoint will be clear from the context).A map T :Md →Md′ is called linear, if T (λA + B) = λT (A) + T (B) ∀A, B ∈ Md, λ ∈ C. Apartfrom the following paragraph with elementary definitions, every map, unless otherwisespecified, will considered to be linear.For A ∈ Md we say A ≥ 0, if A is self-adjoint and all eigenvalues are nonnegative. We writeA > 0 if all eigenvalues are positive. This induces a partial order on the self-adjoint matrices:A ≥ B on A, B ∈ Md if A, B self-adjoint and A − B ≥ 0. In addition, a matrix A ≥ 0 is called astate or a density matrix, if tr(A) = 1, where tr denotes the usual matrix trace. States, althoughmatrices, will usually be denoted by Greek letters.In this thesis, an algebra is not necessarily associative or unital. Given a tensor productA⊗ B oftwo algebras, we denote the partial trace over one of the factors by a subscript, i.e. tr1 denotes thepartial trace overA and tr2 the partial trace over B.Given a linear map T :Md →Md′ , denote by T ∗ the adjoint of the map, defined via 〈A,T (B)〉 =

〈T ∗(A), B〉 (A ∈ Md′ , B ∈ Md), where 〈., .〉 = tr(.†.) defines the usual Hilbert-Schmidt innerproduct onMd′ (Md respectively).Finally, to shorten notation, we sometimes write block-diagonal matrices as diag(A, B, . . .), whereA, B, . . . correspond to the different blocks.

Elementary definitions and lemmata:

First we recall the basic definitions of positive maps:

Definition 2.1. Let T :Md →Md′ be a linear map Then

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• T is called positive, if for all A ≥ 0, T (A) ≥ 0. In particular, a positive map is Hermitian,i.e. T (A)† = T (A†) for all A ∈ Md

• T is called unital, if T (1) = 1

• T is called trace-preserving, if for all A ∈ Md, tr(T (A)) = tr(A).

In chapter 5 we will prove some basic structure theorems for the projections onto fixed pointspaces, involving the notion of decomposable maps. Let us review the definitions here:

Definition 2.2. Let T : Md → Md′ be a linear map, then T is called k-positive for k ∈ N, ifT ⊗ 1k is a positive map. It is called completely positive, if it is d-positive (in fact, one defines Tto be completely positive as being n-positive for all n, but this is equivalent to the definition here,see e.g. [40]).T is called k-copositive, if T = θ ⊗ P, where P is a k-positive map and θ is the transposition map.Likewise, T is called completely copositive, if T is d-copositive.The transposition map in a given basis |ei〉 ∈ Md is given by θ(|ei〉〈e j|) = |e j〉〈ei|. From henceforth,we will always consider the transposition θ to be taken in the standard (computational) basis.

This lets us define decomposable maps:

Definition 2.3. A positive linear map T :Md →Md′ is called decomposable, if it can be writtenin the form

T := T1 + θ T2 (2.1)

where T1 and T2 are completely positive maps and θ is the transposition map. If T is notdecomposable, then it is indecomposable. If T1 and T2 are only k-positive and k-copositiverespectively (for some k < d), then we call the map k-decomposable.

Definition 2.4. A positive linear map T :Md →Md′ is called atomic, if it is not 2-decomposable.

Now we introduce the main objects of interest:

Definition 2.5. Let T :Md →Md be a positive, linear map. Its set of fixed points FT is givenby:

FT := X ∈ Md |T (X) = X (2.2)

Furthermore we define the projector onto the fixed point space via [40]:

T∞ :=∑λ=1

Pλ = limN→∞

1N

N∑n=1

T n (2.3)

12


where λ are eigenvalues of T and Pλ is the spectral projection corresponding to λ and the secondequality holds for trace-preserving maps.In addition, we define the peripheral spectrum of T :

XT := X ∈ Md |T (X) = eiθX, θ ∈ R (2.4)

Again, via [40] we obtain the projection onto the peripheral spectrum:

Tφ :=∑|λ|=1

Pλ (2.5)

Proof. In principle we have to show that equation (2.3) holds. This is easily done by notingthat since T is trace-preserving, every eigenvalue must lie on the complex unit disc, sinceT ∗(X) ≤ ‖X‖∞T ∗(1) = ‖X‖∞ by unitality, hence the spectral radius of both T and T ∗ is % = 1.Considering the linear representation T of T , its Jordan decomposition reads

T =

K∑k=1

(λkPk + Nk) Ndkk = 0, NkPk = PkNk = Nk

and the Pk are mutually orthogonal, i.e. PkPl = δklPk. Since Ndkk = 0 for some dk ∈ N,

limn→∞

1n

n∑i=1

T i = limn→∞

1n

i∑i=1

K∑k=1

λkPk

i

In addition, for λk , 1,∑n

i=1 λik = λ−λn+1

1−λ → 0 for n→ ∞. This gives the quoted equivalence.

We also cite the following lemma:

Lemma 2.6 ([40]). Given T trace-preserving, positive and linear, then there exists a subsequenceof numbers nii ⊂ N s.th.

Tφ = limi→∞

T ni (2.6)

For convenience, we collect a number of simple facts about fixed points of positive maps:

Lemma 2.7 ([40]). For a linear, positive map T :Md(C)→Md(C) we obtain:

• The fixed point space is a †-closed, complex vector space (and hence spanned by itsself-adjoint elements)

• If T is trace-preserving, the fixed point space is spanned by positive elements.

• If T is trace-preserving, then every fixed point has support and range within the supportspace of the fixed point T∞(1).

13

Martin Idel Basic notions of Jordan algebras

Proof. We follow the proofs in [40]:Clearly, the fixed point space is a complex vector space for every linear map. Also, if T ispositive, T is in particular Hermitian preserving, i.e. T (X)† = T (X†), hence the fixed point spaceis †-closed. Therefore, it is also spanned by Hermitian elements since if X is a fixed point, thenX + X† and i(X − X†) are fixed points. Now decompose X = X† into positive and negative part,i.e. X = P+ − P− and tr(P+P−) = 0. In order to show the second part of the lemma, we need tosee that T (P±) = P±. For this, taking Q to be the projection onto the support of P+, we obtain:

tr(P+) = tr(Q(P+ − P−)) = tr(Q(T (P+ − P−))) ≤ tr(T (P+)) = tr(P+)

where we used the orthogonality of P± in the second step, the fact that X is a fixed point of T in thethird and the trace-preserving property of T in the last equation. To see that the inequality holds,we note that positive elements are mapped to positive elements and tr(QT (P+)) ≤ tr(T (P+)).Since we need to have equality obviously, we need tr(QT (P+)) = tr(T (P+)) and tr(QT (P−)) = 0,i.e. T (P+) is supported on Q and tr(QT (P−)) = 0, hence T (P−) is supported orthogonal to Q. Butthen, T preserves the orthogonality of P± and hence T (P±) = P±.For the third part of the lemma, note that for any positive X ∈ Md, we have that X ≤ ‖X‖∞1.Then the positivity of T∞ implies:

T∞(X) ≤ T∞(‖X‖∞1) = ‖X‖∞T∞(1)

hence the support of X has to be contained in the support of T∞(1). This follows from Douglas’lemma, which is a standard result in linear algebra and will not be proven here, since the assertionhere is a bit technical but easy to prove (for a proof and statement of the full lemma, see [7]).For an arbitrary X ∈ Md, we can write it as a linear combination of positive elements P j via thesecond part. Then using that supp(X) ⊂ supp(T∞(1)), we obtain the same assertion immediately.Note that by using that X is a fixed point iff X† is a fixed point and that the range of X equals thesupport of X†, we obtain also that the range of T∞(X) lies within the range of T∞(1).

2.1. Basic notions of Jordan algebras

Let us now recall some basic definitions about Jordan algebras. For a general survey on Jordanalgebras and Jordan operator algebras, see, e.g. [12], [16] or [22].

Definition 2.8. A Jordan algebra is an algebraA over a field F, which contains a multiplicativeinverse of 1 + 1 (in our case we will always work over C or R), whose product ∗ is commutativebut not necessarily associative and fulfills the Jordan identity

(x ∗ y) ∗ (x ∗ x) = x ∗ (y ∗ (x ∗ x)) ∀ x, y ∈ A (2.7)

A Jordan operator algebra is a Jordan algebra over the real or complex numbers, which is also aBanach algebra.

14


Since we will mostly be concerned with finite dimensional matrix algebras, all Jordan algebrasare immediately Jordan operator algebras. However, we want to point out the slight difference,which becomes important in infinite dimensions, where there exists a whole zoo of differentJordan operator algebras corresponding to different topologies (see e.g. [12]).

Definition 2.9. We define the symmetrized matrix product

A ∗ B :=12

(AB + BA) ∀A, B ∈ Cd×d

which makes the set of matricesMd a (complex) Jordan algebra, denoted by (Md, ∗). However,we frequently drop the product in the notation, if it is clear from the context. Moreover, wenote that the set of self-adjoint elements ((Md)h, ∗) is a Jordan algebra in its own right, whenconsidered as a real vector space, since (A ∗ B)† = A† ∗ B∗.

Definition 2.10. LetA be an algebra (either an associative algebra or a Jordan algebra). Thenwe define its centre C to be the set of all elements that are associative and commute with all otherelements of the algebra. More precisely:

C := A ∈ A|A B = B A, A (B C) = (A B) C,∀B,C ∈ A

where is the product of the algebra (either the Jordan product, or the associative product).

The following elements of the structure theory of Jordan algebras will be helpful to us:

Definition 2.11. • A Jordan algebra J over the reals is called formally real if for arbitraryn ∈ N

n∑i=1

X2i , 0 ∀Xi ∈ J , Xi , 0, i = 1, . . . n (2.8)

• A Jordan algebra J is called nondegenerate, if it has no absolute zero divisor except 0 ∈ Jin the sense that for a ∈ J ,

UA : A → J ; X 7→ AXA := A ∗ X ∗ A + X ∗ A2 − A2 ∗ X

it holds that UA = 0 ⇒ A = 0.

• An ideal of a Jordan algebra J is a subspace I ⊂ J s.th. J ∗ I = I = I ∗ J . A Jordanalgebra is called simple, if the only ideals in J are 0 and J , which are trivially ideals.

• A Jordan algebra J is called semi-simple, if it is the direct sum of simple algebras.

• A Jordan algebra J is called special, if it can be embedded into an associative algebraA,i.e. there is an injective map σ : J → A such that

(A ∗ B)σ =12

(AσBσ + BσAσ) ∀A, B ∈ J (2.9)

A Jordan algebra that is not special is called exceptional.

15


The last definition is in fact a representation of a Jordan algebra. We will consider representationsagain in chapter 3. Here, we first give an overview of Jordan algebras and state (without proof) avery useful result from general Jordan theory:

Proposition 2.12 ([16]). A special Jordan algebra is semi-simple if it is nondegenerate. Inparticular, it is semi-simple, if it has a unit.

We do not give a proof of this proposition here, because it is very technical and requires a lotmore definitions and lemmata (e.g. the knowledge of Jacobson-radicals).Since we will only consider Jordan algebras, which are subalgebras of matrix algebras, fromnow on every Jordan algebra will be assumed to be special. In this case, we have two productsto deal with, the symmetrized Jordan product, which will always be denoted by ∗ and the usualassociative matrix product, which will be denoted by · or omitted for readability.It might be interesting to note that this restriction is not very profound, since when workingover fields with characteristic zero there is only one exceptional Jordan algebra, the so-calledAlbert algebra (see e.g. [22]). This is essentially the reason why the original program for thedefinition of Jordan algebras by Jordan (see [17]), was abandoned. Jordan (et al.) hoped tothereby create algebras of purely self-adjoint elements that could serve as operator algebras forquantum mechanics without having to refer to some bigger, associative algebra, which seemedfutile, once it was realized that there were essentially no exceptional Jordan algebras.As a further recollection of classical results about Jordan algebras, we obtain (the proofs of thesefacts can be found in [16] or [15], the classification theorem for formally real Jordan algebraswas first derived in the founding paper [17]):

Theorem 2.13 (Classification of formally real (special) Jordan algebras). Every formally realJordan algebra is unital, nondegenerate and semi-simple. Furthermore, the simple formally realJordan algebras in finite dimensions can be classified:

• (Md(R))h the self-adjoint d × d-matrices over the reals.

• (Md(C))h the self-adjoint d × d-matrices over the complex field.

• (Md(H))h the self-adjoint d × d-matrices over the quaternions.

• The spin factors or Jordan algebras of Clifford type, given by R1 + Rn, where the Jordanproduct is defined via: ∀(α1 + v), (β1 + w) ∈ R1 + Rn

(α1 + v) ∗ (β1 + w) = (αβ + 〈v,w〉)1 + βw + αv

where 〈., .〉 denotes the inner product on Rn. We will denote a spin factor of dimension k + 1byVk

This implies that a formally real Jordan algebra is either a spin factor or an algebra of Hermitianelements of a matrix algebra over some finite-dimensional associative division algebra (byFrobenius theorem, the associative division algebras are exactly R,C or H).Let us furthermore define a spin system:

16


Definition 2.14. A spin-system is a set of n elements ei ∈ Rn such that ei ∗ e j = δi j1. A

spin-system with n elements together with the identity will be denoted by Vn to allude to the factthat it generates a spin factorVn.

Before continuing, we will give another characterization of spin factors:

Lemma 2.15 ([12], 6.1.5.). Every spin factor is isomorphic to a spin factor generated by theidentity and a spin-system. Furthermore, any spin-system of smaller size can be extended to agenerating spin-system.

Proof. Given our definition of a spin factor, any orthonormal basis of Rn is a spin-system andcan be extended to a basis by the basis-extension theorem.

A notion that will turn out to be essential for us and which is tightly linked to spin factors (as thelemma below shows) is reversibilty of Jordan algebras:

Definition 2.16. A Jordan algebra J ⊂ Md for some d ∈ N is called reversible, if for everyn ∈ N we have:

a1 . . . an + an . . . a1 ∈ J ∀ai ∈ J , i = 1, . . . n (2.10)

Otherwise, it is called irreversible.

We collect the following easy observations:

Lemma 2.17. For any Jordan algebra J , we have

12

(abc + cba) = (a ∗ b) ∗ c + (b ∗ c) ∗ a − (a ∗ c) ∗ b ∀a, b, c ∈ J

In fact, one can show that a Jordan algebra is irreversible, iff the definition for reversibility failsfor n = 4, but since we are not going to use this fact, we will not give it as a lemma.

Corollary 2.18. The Jordan algebra of self-adjoint operators (Md)h ⊂ Md (or any real subalge-bra thereof) is a formally real Jordan algebra, which is in particular nondegenerate.

Proof. Consider the case (Md)h. This is a real algebra, which is closed under the Jordanproduct, hence it is a real Jordan algebra. We need to see that it is also formally real, i.e.we for any set of nonzero Ai ∈ (Md)h, i = 1, . . . n (n ∈ N arbitrary)

∑ni=1 A2

i , 0. But thisis true, since in particular A2

i > 0 if Ai , 0. Furthermore, for 0 , A ∈ (Md)h considerUA(B) = A ∗ B ∗ A + B ∗ A2 − A2 ∗ B = ABA, then ABA = 0 for all B ∈ Md if and only if A = 0,because choosing B = A, we obtain UA(A) = A3 , 0 if A , 0, hence (Md)h is nondegenerate.If we consider subalgebras of the self-adjoint matrices, the properties are obtained via the sameproof.

Furthermore, let us introduce the concept of an enveloping algebra:

17

Martin Idel Fixed point spaces of positive maps

Definition 2.19 ([16]). Given a finite dimensional Jordan algebra J its enveloping algebra S(J)is the associative algebra generated by the elements of J .

If J is a Jordan-subalgebra of (Md, ∗), then S(J) is the smallest associative subalgebra ofMd

that contains J . If in addition, J is a †-closed algebra, S(J) is the smallest C*-subalgebra ofMd that contains J .As a last important concept, we will introduce the free-algebras and prove their universal proper-ties:

Definition 2.20 ([16, 12]). The free associative algebra FA(a1, . . . , an) is the associative algebragiven by all the complex linear combinations of words over a1, . . . , an (the operation is justconcatenation of letters), where the empty word is denoted by 1 and is a unity of the algebra.The free reversible Jordan algebra FS(a1, . . . , an) is the subalgebra of FA(a1, . . . an) consisting ofall the words of the form ai1 . . . aim + aim . . . ai1 with i j ∈ 1, . . . , n for all j = 1, . . . ,m and theirlinear combinations. Again, the empty word is denoted by 1.

Proposition 2.21. We have the following universal properties:

• [18] (as free module): Given a unital, associative algebraA with elements x1, . . . , xn ∈ A

for some n ∈ N, there exists a unique homomorphism h : FA(a1, . . . , an)→ A with 1 7→ 1and ai 7→ xi for all i ∈ 1, . . . , n.

• [16],[12]: Given a unital Jordan algebra J with elements x1, . . . , xn ∈ A for some n ∈ N,there exists a unique Jordan-homomorphism h : FS(a1, . . . , an) → J with 1 7→ 1 andai 7→ xi for all i ∈ 1, . . . , n.

Proof. The first part is clear, since we can define h via h(ai) = xi for all i ∈ 1, . . . , n andextend by linearity and homomorphism. Since the extension is unique, h is unique. The secondpart follows from restricting h defined in the first part, since the homomorphism h restricts to aJordan-homomorphism on FS.

2.2. Fixed point spaces of positive maps

The goal of this section is to analyze the structure of fixed points of positive, linear maps. Forconvenience, let us review the Schwarz-inequalities:

Definition 2.22. A positive map T : Md → Md is a Schwarz-map if it satisfies the Schwarz-inequality:

T (A†A) ≥ T (A)†T (A) (2.11)

and it is an anti-Schwarz-map, if it satisfies the anti-Schwarz-inequality:

T (A†A) ≥ T (A)T (A)† (2.12)

18


From e.g. [40] it is well-known that any 2-positive map is a Schwarz-map and that not everypositive map is a Schwarz-map. Since we will need the fact that 2-positive maps are Schwarz-maps and 2-copositive maps are anti-Schwarz-maps, we will prove it here:

Lemma 2.23 ([40]). Let T : Md → Md be a 2-positive map. Then it satisfies the Schwarz-inequality. Likewise, if T is 2-copositive, then it satisfies the anti-Schwarz-inequality.

Proof. It satisfies to prove the first assumption, since the second assumption follows by usingthat θ T is 2-positive and using that θ is an anti-automorphism.To prove the first assumption, we follow the proof in [40]. Let A ∈ Md. Then consider C := (A,1),then C†C is a positive semidefinite matrix. In particular:

C†C =

(A†A A†

A 1

)Since T is 2-positive, we obtain C†C ≥ 0 ⇒ (id2 ⊗ T2)(C†C) ≥ 0. But using [1], theorem 1.3.3.,this implies T (A†)T (A) ≥ T (A†A).

However, every positive, unital map satisfies a similar inequality for the Jordan product:

Lemma 2.24 (Jordan-Schwarz-inequality). 1 Let T :Md →Md be a unital positive map. Thenfor all A ∈ Md:

T (A† ∗ A) ≥ T (A†) ∗ T (A)

Proof. For Hermitian matrices X ∈ (Md)h Kadison’s inequality reads:

T (X2) ≥ T (X)2

Now for arbitrary A ∈ Md, X1 = A + A† and X2 = i(A − A†) are Hermitian and we obtain:

T (A2 + A†2) + 2 · T (A† ∗ A) ≥ (T (A) + T (A†))2

−T (A2 + A†2) + 2 · T (A† ∗ A) ≥ −(T (A) − T (A†))2

Adding the two inequalities and dividing by four gives the inequality.

Lemma 2.25. Given X ∈ Md, s.th. for a positive map it holds

T (X† ∗ X) = T (X†) ∗ T (X)

then

T (B† ∗ X) = T (B†) ∗ T (X) ∀B ∈ Md

1The proofs of this and the following lemmas are probably well-known, they also appear in [8] and are supposedlyknown from Boise. This version was given by M.M. Wolf through private communication

19


Proof. Let X ∈ Md fulfill equality in the Jordan-Schwarz-inequality. We apply the first lemma toA := tX + B with t ∈ R. We calculate:

T (A† ∗ A) ≥ T (A†) ∗ T (A)

⇔ t2T (X† ∗ X) +12

tT (X†B + B†X + XB† + BX†) + T (B† ∗ B) − T (B†) ∗ T (B)

≥ t2T (X†) ∗ T (X) +12

t(T (X†)T (B) + T (B†)T (X)

+ T (X)T (B†) + T (B)T (X†))

⇔12

tT (X†B+B†X + XB† + BX†) + T (B† ∗ B) − T (B†) ∗ T (B)

≥12

t(T (X†)T (B) + T (B†)T (X) + T (X)T (B†) + T (B)T (X†))

We repeat the same calculations for A = itX + B and obtain:

12

itT (−X†B+B†X + XB† − BX†) + T (B† ∗ B) − T (B†) ∗ T (B)

≥ +12

it(T (−X†)T (B) + T (B†)T (X) + T (X)T (B†) + T (B)T (−X†))

Now the above equation holds for all t ∈ R. Hence the linear terms must vanish, since:

12

tT (X†B + B†X + XB† + BX†) + T (B† ∗ B) (2.13)

≥12

t(T (X†)T (B) + T (B†)T (X) + T (X)T (B†) + T (B)T (X†)) + T (B†) ∗ T (B) (2.14)

⇔ T (B† ∗ B) − T (B†) ∗ T (B) +12

t[T (X†B + B†X + XB† + BX†) (2.15)

− (T (X†)T (B) + T (B†)T (X) + T (X)T (B†) + T (B)T (X†))] ≥ 0 (2.16)

Now if the linear term in (2.16) does not vanish, we can always find some t ∈ R which violatesthe above equation thus showing that the linear term has to vanish. But then, we have

T (X†B+B†X + XB† + BX†)

− T (X†)T (B) + T (B†)T (X) + T (X)T (B†) + T (B)T (X†) = 0

T (−X†B+B†X + XB† − BX†)

− T (−X†)T (B) + T (B†)T (X) + T (X)T (B†) + T (B)T (−X†) = 0

Adding both equations and dividing by four, we obtain the desired result:

12

(T (B†X + XB†)) =12

(T (B†)T (X) + T (X)T (B†))

20


With our knowledge of Jordan algebras, we obtain the following theorem, which extends atheorem in [40] (chapter 6, theorem 12) for Schwarz-maps and which appears also in [8] (cor. 1.6,apparently pointed out by Alain Connes), albeit in a more general form and proved differently:

Theorem 2.26. Let T ∗ :Md →Md be a unital, positive map such that T has a full-rank fixedpoint (i.e. T is similar to a unital map). Then FT ∗ is a semi-simple, unital Jordan algebra, whoseself-adjoint part is a formally real, semi-simple Jordan algebra. Furthermore, if T satisfies theSchwarz-inequality, the Jordan algebra is in fact a C*-algebra.

Proof. The proof basically follows the proof of the C*-case presented in [40], replacing theSchwarz-inequality by the Jordan-Schwarz-inequality:Let ρ ∈ Md(C) be a full-rank positive fixed point of T . Now consider a fixed point A ∈ FT ∗ , i.e.T ∗(A) = A. By lemma 2.24, it follows T ∗(A† ∗ A) ≥ T ∗(A)† ∗ T ∗(A) = A† ∗ A. We also get:

tr(T ∗(A† ∗ A)†ρ) = tr((A† ∗ A)†T (ρ)) = tr((A† ∗ A)†ρ) = tr((T ∗(A)† ∗ T ∗(A))†ρ)

This implies:

tr((T ∗(A† ∗ A) − T ∗(A)† ∗ T ∗(A))†ρ) = 0

Now by the Jordan-Schwarz-inequality, T ∗(A† ∗ A) − T ∗(A)† ∗ T (A) − T (A)† ∗ T (A) ≥ 0, henceequality must hold in the Jordan-Schwarz-inequality for all A ∈ FT ∗ , since ρ is positive definite.Using lemma 2.25, we obtain that T ∗(A ∗ B) = T ∗(A) ∗ T ∗(B) = A ∗ B for all A, B ∈ FT ∗ , hencethe fixed point space must be a complex Jordan algebra, as we already know it to be a complexvector space from lemma 2.7. Since the Jordan product of two self-adjoint matrices is stillself-adjoint and the self-adjoint part of the fixed point algebra is naturally a real vector space,we obtain that it is also a Jordan algebra, which is formally real (and unital) by construction.Consequently, by corollary 2.18, the Jordan algebra is semi-simple, thus also the complex Jordanalgebra generated by its self-adjoint part, which corresponds to the fixed point space. If theSchwarz-inequality holds instead of the Jordan-Schwarz-inequality, the proof holds also with ·instead of the ∗-product, hence we obtain a *-algebra.

We can also drop the condition that T has a full rank fixed point:

Corollary 2.27. Let T : Md → Md be a trace-preserving, positive map, where Q is theprojection onto the support of the maximum rank fixed point of T (see lemma 2.7). Then thefollowing set is a Jordan algebra:

Y ∈ QMd(C)Q |QT ∗(Y)Q = Y (2.17)

Again, if T fulfills the Schwarz-inequality, this set is a *-algebra and not only a Jordan algebra.

Proof. Again, we follow the proof in [40]:Note that for T a trace-preserving, positive map, in lemma 2.7 we have stated that every fixed

21


point has support in supp(T∞(1)). The idea of the proof will be to construct a map T acting onlyon supp(T∞(1)), such that T and T act in the same way on fixed points. Then T has a full-rankfixed point. We then have to connect the fixed point space of T ∗ with the fixed point space of T ∗

and if T is also positive and trace-preserving, then we can use the theorem for T , inferring FT ∗

from FT ∗ .To this end, let V : H → Cd be an isometry which embedsH into C such that Q := VV† is theprojection onto supp(T∞), then T : B(H)→ B(H) with T (X) = V†T (VXV†)V is a positive map,since T is a composition of positive and completely positive maps and thus positive. We need toshow that the map is trace-preserving. To this end, decompose VXV† = X ⊕ 0 with support andrange within the range of Q (0 acts on the orthogonal complement ofH , or put differently, writeV : |ψ〉 7→ |ψ〉 ⊕ 0). We want to show that then T (X ⊕ 0) = T (X) ⊕ 0. Then also the set of fixedpoints of T equals the set of fixed points of T .First consider positive X⊕0 and observe that X⊕0 ≤ cT∞(1) for some constant c > 0, since X⊕0has support and range inside the support and range of T∞(1). Also, by definition of the projection,T∞(1) ≤ c′Q for some constant c′ > 0 and since T∞(1) is a fixed point of T , we obtain thatT (X ⊕ 0) ≤ c′cQ, i.e. especially T (X ⊕ 0) has range and support inH , i.e. T (X ⊕ 0) = T (X) ⊕ 0.Therefore, T is trace-preserving, since T is trace-preserving.Since all fixed points have support (and range) in H (again, via lemma 2.7), hence are of theform X ⊕ 0, which implies FT = FT ⊕ 0 and T has a full-rank fixed point.Now suppose we knew that if A ∈ FT ∗ , then V†AV ∈ FT ∗ and X ∈ FT ∗ iff V†T ∗(VXV∗)V = X.But then, the set given in the corollary is exactly FT ∗ , which is a Jordan algebra by the theorem(or a *-algebra if T fulfills the Schwarz inequality).It remains to prove that (1) if A ∈ FT ∗ , then V†AV ∈ FT ∗ and (2) X ∈ FT ∗ iff V†T ∗(VXV†)V = X.To this end, note that T ∗(V†AV) = V†T ∗(QAQ)V (this gives (2) once we know (1)). LetC ∈ B(H) be arbitrary, note that tr(T (VCV†)QAQ) = tr(T (VCV†)A), since VCV† ≤ const Q forsome constant by definition of V and tr((1 − Q)T (Q)) = 0. Then we obtain for arbitrary C:

tr(C†T ∗(V†AV)) = tr(T (VC†V†)QAQ) = tr(T (VC†V†)A) = tr(C†V†AV)

But then V†AV ∈ FT ∗ .

One of the most important observations in this section is the fact that the fixed point space FT ∗

of a positive, trace-preserving map T is a semi-simple Jordan algebra. In order to get deeperinsights, we now study positive projections onto Jordan algebras.

22

3. Representation theory of Jordan algebras

In this chapter, we want to study finite dimensional representations (i.e. Jordan homomorphismsof abstractly defined Jordan algebras into complex matrices) of Jordan algebras. Since in principle,any Jordan subalgebra of the complex matricesMd could occur as a fixed point space of somepositive map, we need to consider all possible Jordan subalgebras. In consequence, it will behelpful to get to know more about their structure. The representation theory of Jordan algebraswill be the fundamental ingredient to constructing projections onto Jordan algebras, which willbe done in chapter 4.The problem of considering all Jordan algebras J that are subalgebras ofMd is equivalent to theproblem of classifying faithful (i.e. injective) representations of Jordan algebras. It is crucial tonote that when considering representations into the set of complex matrices, we actually have todeal with the complex Jordan algebras and not the formally real Jordan algebras, since these arethe actual fixed-point algebras (e.g. we have to deal with the set of symmetric elements ratherthan only the set of real symmetric matrices). However, there is a simple connection, which willbe exploited in order to still use our classification in theorem 2.13.Let us first give a general overview of the main result of this chapter and outline the thoughtsleading to this result. To this end, we give the basic definition of a representation:

Definition 3.1. A representation of a finite dimensional Jordan algebra J (with Jordan product·) into a finite dimensional associative algebraA Md (with symmetric product ∗) is a linearmap σ : J → A such that

(A · B)σ = Aσ ∗ Bσ ∀A, B ∈ J (3.1)

Such maps are also called associative specializations, alluding to the fact that such a Jordanalgebra is by definition a special Jordan algebra.In addition, we call a representation faithful (or an embedding), if it is injective and we introducethe notation Jσ to denote the represented subalgebra (i.e. the image of the representation), if it isnecessary to distinguish between the abstract algebra and a specially chosen representation.Likewise, considering the envelope S(J) of a representation of a Jordan algebra J , we specifythe representation σ via S(J , σ) if necessary.

Note that there are also different representations, called multiplicative representations, see [16].

Definition 3.2. Let I,J be two Jordan algebras (with Jordan product ∗), then a map T : I → Jis a Jordan homomorphism, if it is linear and respects the Jordan structure, i.e.

T (A ∗ B) = T (A) ∗ T (B)

23


With this definition in mind, we state the main result, the proof of which will be obtained insection 3.4

Theorem 3.3 (Classification of complex Jordan algebras). Given a unital Jordan subalgebra JofMd, it is (up to isomorphism) of the following form:

J =

K1⊕i=1

Mdi ⊗ 1mi

⊕ K2⊕

i=1

JCdi

⊕ K3⊕

i=1

JTd′i⊗ 1m′i

⊕

K4⊕i=1

JHd′′i⊗ 1m′′i

⊕ K5⊕

i=1

Jspinfullki⊗ 1m′′′i

⊕ K6⊕

i=1

Jspinsumki

(3.2)

where mi,m′i ,m′′i ,m

′′′i ∈ N and the unknown factors are given by

1. d′i ∈ N, JTd′i

= S (Md′i (R) + iMd′i (R))S † for some unitary S .

2. d′′i ∈ 2N, d′′i ≥ 6, and set d′′i = 2ki, thenJHd′′i⊂ Md′′i andJH

d′′i= S (Mki(H)σ+iMki(H)σ)S †

for some unitary S ∈ Md′′i . Here, (Mki(H)σ + iMki(H)σ) is the complex extension of theembedding σ of the Jordan algebra of Hermitian quaternions in proposition 3.6 such thatits enveloping algebra isMd′′i .

3. di ∈ N, di ≥ 3, JCdi

= U diag(A, AT )U† with unitary U denote all embeddings of Jordan

algebra generated by the complex Hermitian matrices of dimension di, where the envelopeis isomorphic to the universal envelope and A = A⊗1mi ⊂ Mdimi

and A = A⊗1mi ⊂ Mdimi

for all A ∈ Mdi(C) and some mi, mi ∈ N.

4. Jspinfullki

with 4 ≤ ki = 2ni or 4 ≤ ki = 2ni + 1 is a spin factor of dimension ki + 1 andenveloping algebraM2ni Its embedding is then isomorphic to the universal embeddinggiven in proposition 3.7 for ki = 2ni or isomorphic to the embedding given in proposition3.8 for ki = 2ni + 1.

5. Jspinsumki

with 4 ≤ ki = 2ni − 1 odd is a spin factor of dimension ki + 1. Then eitherJspinsum

ki= S (Vki)

σ0S †, where σ0 is the universal embedding in proposition 3.7 withenvelopeM2ni−1 ⊗Mmi ⊕M2ni−1 ⊗Mmi and S ∈ M2ni−1mi+2ni−1mi

a unitary.

The decomposition given here is unique up to isomorphism.

We have here a very explicit form of the possible fixed point spaces. Using this structure, one cangive similarly explicit constructions of projections onto the Jordan algebras, which can then becharacterized by inspection. In addition, it will be very important to talk about the reversibilityof the given representation. Reversibility is a feature of the representation, since it requires asurrounding associative algebra for its definition (see definition 2.16). We get the followingproposition, which slightly extends a result in [12] (Theorem 6.2.6):

24


Proposition 3.4 ([12]). Every embedding of a Jordan algebra that is not a spin factor is reversible.In addition, spin factors of dimension n = 3, 4 are reversible, while every spin factor of dimensionn ≥ 7 and n = 5 is irreversible. The case n = 6 is special. Here, every embedding, which is*-isomorphic to the universal embedding is irreversible, while every embedding with a simpleenvelope is reversible.Spin factors of dimension n = 6 also correspond to embeddings ofM2(H) and there is exactlyone irreversible embedding of this spin factor.

The proposition tells us that basically, all embeddings of Hermitian matrices are reversible withexactly one exception (a curious embedding ofM2(H)), while spin factors are in general notreversible, unless the dimension is too small.Now let us turn to an outline of the strategy for proving the classification theorem. The mainidea of the proof is to split the question into two different problems: the representation theory ofC*-algebras (which is given by the Artin-Wedderburn theorem, see theorem 3.23) and so-calleduniversal embeddings, which are (intuitively) faithful representations of a given Jordan algebraJ ⊂ Md such that its enveloping algebra S(J) has maximal dimension.Step 1: We first construct embeddings for every type of formally real Jordan algebra in section3.1. These embeddings will later turn out to be universal or the universal embedding can easilybe constructed from these embeddings.Step 2: In section 3.2 we will introduce universal embeddings, prove their existence and clarify,why, given the representation theory of C*-algebras, it suffices to consider these representations.The main point is that any (special) Jordan algebra has an enveloping C*-algebra, hence we needto consider the possible embeddings of enveloping algebras (i.e. C*-algebras) intoMd and theembedding of Jordan algebras into their envelopes. The universality of the universal envelopewill ensure that we find all possible envelopes. For reference, we refer to e.g. [15] and [16]Step 3: Having this in mind, in section 3.3 we present the calculations of [15], where the universalenveloping algebras and the universal embeddings of all complex Jordan algebras are computed,after having seen that there is a one-to-one correspondence between formally real Jordan algebrasand the self-adjoint part of complex Jordan algebras. We will see that the universal envelopes arealways either simple associative or a sum of two simple associative algebras.Step 4: Finally, we can put things together in section 3.4. We know all possible envelopes and allpossible embeddings of envelopes. Furthermore, we know that there are automorphisms betweenall possible universal embeddings and we can extend this result to other embeddings. Henceit remains to find embeddings for each case and to consider the automorphisms between them.With the help of the Skolem-Noether theorem 3.15, we can prove that the automorphisms aremostly inner automorphisms (hence automorphisms ofMd), which gives us the structure as inour characterization theorem with all different isomorphic embeddings.This last section and hence the explicit description of finite dimensional Jordan algebras ofthe main theorem seem to be well-known, as they follow rather straight-forwardly from thecalculation of the universal algebras, but (to our knowledge) have never been written down inexplicit form.

25

Martin Idel Associative specializations and embeddings of formally real algebras

3.1. Associative specializations and embeddings of formallyreal algebras

We now construct embeddings of all formally real algebras. Our later discussion will crucially relyon these faithful representations. Embeddings of the self-adjoint, formally real Jordan algebrasJh intoMd(C) for Jh = (Md(R))h and Jh = (Md(C))h are obvious (inclusion induced). Thetwo non-trivial cases are the quaternionic Hermitian matrices and the spin factors. Let us startwith the quaternionic Hermitian matrices.Recall that quaternion numbers are given as a ∈ H, then a = w + ix + jy + kz where i2 = j2 =

k2 = i jk = −1. Hence we can also write a = (w + ix) + (y + iz) j as a sum of two complex matrices.The idea is to find an embedding of j, just as one can embed the complex numbers into the reals.To this end, consider first the standard embedding of H intoM2(C) via:

w + ix + jy + kz 7→(

w + ix y + iz−y + iz w − ix

)Complex conjugation for quaternions is then just given by complex conjugation of the corre-sponding matrix, or else by w + ix + jy + kz = w− ix− jy− kz. This gives us a hint how to definean embedding of quaternion Hermitian matrices, where a quaternion matrix A ∈ Md(H) is calledHermitian, if A† = A, where † denotes transposition and quaternionic complex conjugation:

Lemma 3.5. Let A ∈ Md(H) be Hermitian, then we can write

A = W + iX + jY + kZ

where W is real symmetric and X,Y,Z are real and skew-symmetric.

Then define:

Proposition 3.6. Given the decomposition above, for every d ∈ N, we have an embedding of thequaternionic Hermitian matrices via:

σ :(Md(H))h → (M2d(C))h

A = W + iX + jY + kZ 7→(

W + iX Y + iZ−Y + iZ W − iX

)(3.3)

The corresponding complex Jordan algebra J = (Md(H))σh + i(Md(H))σh can equally be charac-terized as the set of matrices A ∈ M2d(C) such that either

A ∈ J ⇒ A =

(A11 A12A21 A22

)where we have the following conditions:

A11 = AT22, AT

12 = −A12, AT21 = −A21 (3.4)

26


or A′ = A, where′ :M2d →M2d

A 7→ Q−1AT Q(3.5)

with Q :=(

0 1

−1 0

).

Proof. We first note that we have:

σ :(Md(H))h → (M2d(C))h

A = W + iX + jY + kZ 7→(

W + iX Y + iZ−Y + iZ W − iX

)By the symmetry-properties, W, iX, iZ are self-adjoint, while Y is skew-Hermitian, hence theembedding is well-defined. A quick calculation shows that it also respects the Jordan product,hence is a representation of the Jordan algebra.Now consider the complex Jordan algebraJ := Jh + iJh, whereJh is the constructed embeddingof the quaternionic Hermitian matrices. We immediately have a decomposition as above:

A ∈ J ⇒ A =

(A11 A12A21 A22

)This induces the following conditions:

A11 = AT22 (3.6)

AT12 = −A12 (3.7)

AT21 = −A21 (3.8)

since in the Hermitian case we have (W + iX)T = W − iX, as W was symmetric and X wasskew-symmetric (condition (3.6)) and since Y + iZ were skew-symmetric (condition (3.7) and(3.8)). We claim that these are all the conditions on Ai j.For A11 and A22 there cannot be any more conditions, since W, X, iW, iX spanMd (as the realsymmetric and real and skew-symmetric matrices span all real matrices).For A21 and A12 the condition of being skew-symmetric is obviously necessary, but one might notexpect that A21 and A12 are independent. To see this, it suffices to construct all matrices A ∈ Md

such that

A =

(0 A120 0

)(by symmetry, we also get the matrices where only A21 is nonzero). Of course, A12 needs to beskew-symmetric. This implies:(

0 A120 0

)=

(0 R + iI0 0

)=

12

((0 R + iI

−R + iI 0

)+ i

(0 I − iR

−I − iR 0

))

27


where R, I are real skew-symmetric matrices. Then the last expression contains a linear combina-tion of matrices in Jh. Hence there are no further conditions on A12 (A21). This proves the claimthat indeed, the conditions in equation (3.4) are complete.Now consider the involution

′ :M2d →M2d

A 7→ Q−1AT Q

We claim that J are exactly the matrices A ∈ M2d such that A′ = A. In the notation above:

A′ = Q−1AT Q =

(AT

22 −AT12

−AT21 AT

11

)Comparison gives the conditions as stated in equation (3.4).

Now we construct an embedding of spin factors. It suffices, to construct an embedding of thespin-systems and extend this construction to the linear span. Here, we follow [12]: Let σ1, σ2, σ3be the familiar Pauli-matrices in the following order:

σ1 =

(0 11 0

), σ2 =

(0 i−i 0

), σ3 =

(1 00 −1

)(3.9)

These are self-adjoint and will form the basis of our construction. Furthermore, identifyMn(C)⊗Mm(C) =Mnm(C) =Mm(Mn(C)) via:

A ⊗ 1m = diag(A, . . . , A) ∈ Mm(Mn(C)) A ∈ Mn

1n ⊗ B = (Bi j1n) ∈ Mm(Mn(C)) B ∈ Mm

We apply this construction toM2 and obtain a chainM2 →M22 →M23 , etc.

Proposition 3.7 ([12]). Consider the following representation of a spin-system Vk intoM2n+1

with k = 2n + 2:

S 1 = σ3 ⊗ 1n−1

S 2 = σ1 ⊗ 1n−1

S 3 = σ2 ⊗ σ3 ⊗ 1n−2

S 4 = σ2 ⊗ σ1 ⊗ 1n−2

...

S 2n+1 = σn2 ⊗ σ3

S 2n+2 = σn2 ⊗ σ1

where σni denotes the n-fold tensor product. For k = 2n + 1 or k = 2n + 2, the sets S 1, . . . , S 2n+1

and S 1, . . . , S 2n+2 extend to an embedding of the spin factorVk intoM2n+1 .

28

Martin Idel Reduction of the representation theory to universal algebras

Proof. Taking the embeddings of the tensor products above, we obtain that S k ∈ M2n if k ≤ 2n.Using σi, σ j = 2δi j1 the usual Clifford-algebra, we obtain that S i ∗ S j = δi j1, hence we have aspin-system and by linear extension a spin factor of dimension k + 1. This gives us an embeddingof a spin-system of arbitrary finite dimension as claimed. The fact that the enveloping algebrasare as stated will be given later on in lemma 3.22.

At this point let us remark that the infinite dimensional case results in the usual CAR algebra,which we do not consider here ([12]).

Proposition 3.8. Consider k = 2n − 1, then proposition 3.7 gives an embedding ofVk intoM2n

with envelope S(Vk) =M2n−1 ⊕M2n−1 .There is also an embedding σ1 : Vk →M2n−1 by using an embedding of S 1, . . . , S 2n intoM2n−1

as a spin factor of type k = 2n and add S 2n+1 := ε2 (S 1 . . . S 2n + S 2n . . . S 1), where ε = 1, i such

that S 2n+1 is self adjoint.

Proof. We just need to prove that the given map is an embedding of a spin-system. To this end,note that S i ∗ S j = 0 for i , j and S 2

i = 1 for i, j ∈ 1, . . . , 2n since we have an embedding ofthe spin system V2n Via the anticommutation relations of the S n, since S 2n+1 is an even linearcombination of S i, S 2n+1 ∗ S i = 0 for all i ∈ 1, . . . , 2n. Furthermore, S 2

2n+1 = 1, hence we havean embedded spin system, which extends to an embedding of the spin factorVk as claimed.

3.2. Reduction of the representation theory to universalalgebras

Definition 3.9 ([15]). A representation A 7→ Aσ0 of a Jordan algebra J is called a universalembedding and its enveloping algebraU := S(J) is called a universal enveloping algebra for J ,if the correspondence Aσ0 7→ Aσ determined by an arbitrary representation σ can be extended toa (algebra)-homomorphism of the enveloping algebraU onto the enveloping algebra S(J , σ).

We can visualize this definition in a commutative diagram, which makes it easier to remember:

J S(J , σ)

S(J , σ0)

∃ h hom.σ0

σ

From now on, every homomorphism and isomorphism that is not called a Jordan homomorphism(isomorphism) is an algebra homomorphism of the associative algebras. We are interested inalgebra-homomorphisms, because, since any Jordan algebra is in particular a linear subspace,linear homomorphisms and their extensions on the level of vector spaces are pretty much mean-ingless.First of all, we need the following abstract existence theorem for universal algebras:

29


Proposition 3.10 ([15, 16]). Given a finite-dimensional, special Jordan algebra J , it has auniversal embedding.

Proof. We follow here [12] (they only consider the real case; the general case is the same, adifferent source is [16]). Consider the special Jordan algebra J and define:

T (J) :=∞⊕

i=1

⊗n

J

Then T (J) is an associative algebra with ⊗ as multiplication (associative, since the tensor productis associative). Now let I be the ideal generated by A ⊗ B + B ⊗ A − 2A ∗ B for A, B ∈ J anddefineU := T (J)/I and claim that it is a universal envelope.Consider an embedding σ of J into an associative algebraA, i.e. a linear map

σ : J → A

A 7→ Aσ

Then, via the universal property of the tensor product (see e.g. [18]), this extends to a homomor-phism

h : T (J)→ A

h(A1 ⊗ . . . ⊗ An) = Aσ1 . . . Aσn

Now since

h(A ⊗ B + B ⊗ A − 2A ∗ B) = AσBσ + BσAσ − 2(A ∗ B)σ = 0

the homomorphism descends to a homomorphism h : U → A. Since the embedding wasarbitrary,U fulfills the universal property.

Furthermore, we will need the (intuitive) theorem about direct sums of algebras, which lets usfocus only on the simple algebras:

Proposition 3.11 ([15]). Given a unital Jordan algebra J such that J = J1 ⊕ J2, then itsuniversal algebra U also decomposes as U = U1 ⊕ U2 such that Ui are generated by therespective Ji.

Proof. The proof follows along [15] (theorem 2.1 and 2.2). Let J = J1 ⊕ J2 be a unitalsubalgebra of some associative algebra A. Then, by unitality, we have E1 ∈ J1 which is anidempotent acting as an identity forJ1 (with respect to the Jordan product), i.e. E1∗A = A∗E1 = Afor all A ∈ J1. Likewise, we have and identity E2 ∈ J2. Now E1E2 = E2E1 = 0 via the directsum structure. Furthermore, for all A ∈ J :

A = E1 ∗ (E1 ∗ A) = E1 ∗ (E1A + AE1)/2 = (E1A + AE1)/4 + E1AE1/2

30


and since E1 ∗ A = A, (E1A + AE1)/2 = E1AE1(= A) hence E1A = AE1. Then, E1 is an identityalso for S(J1). Likewise, E2 is an identity for S(J2) as an associative algebra.On the other hand, for A ∈ J2, E1 ∗ A = 0, ⇒ E2

1A = −E1AE1 = AE21, hence E1A = AE1 = 0.

This means that E1 and E2 are orthogonal identities for S(J1) and S(J2), hence the two algebrasare a direct sum, i.e. S(J1 ⊕J2) = S(J1)⊕S(J2). In particular, this implies that the envelopingalgebra of a direct sum is a direct sum and we only need to see that the Ui are universalfor Ji. To this end, consider an embedding σ into an associative algebra with envelope A,such that A1 + A2 7→ Aσ1 + A2 (for all A1,2 ∈ J1,2). Then the enveloping algebra of thisrepresentation isA⊕U2 and by the universal property, the embedding extends to a homomorphismh : U1 ⊕U2 → A⊕U2 of the associative algebras. The restriction to the first summandA is anextension of the representation σ to a homomorphism h|U1 : U1 → A. Since σ is arbitrary,U1fulfills the universal property for J1, hence is universal. An analogous argument gives thatU2 isuniversal.

We derive a few consequences, which illuminate the way to a characterization of representationsof Jordan algebras:

Lemma 3.12. Given a Jordan algebraJ with its universal embedding σ0 and universal enveloping

algebra U, any correspondence π : Jσ0 → Jσ, Aσ0 → Aσ between the universal embeddingand another representation σ extends to a surjective *-homomorphism between the envelopingalgebras π : U → S(J , σ), which is an isomorphism if the enveloping algebras are simple.

Proof. We have π : Jσ0 → Jσ is a Jordan-*-isomorphism. By the universal property of π, themap extends to a map π between the envelopes, which is a *-homomorphism. Furthermore, theenveloping algebras are necessarily *-algebras.In the general case, since we already have a (*)-homomorphism, we only need to show surjectivityof π, but this follows from the fact that π is onto, since, given a generating system A1, . . . , Am ⊂

J , such that (Aσ0i1

) . . . (Aσ0in

) with i1, . . . , in ∈ 1, . . . ,m generate the enveloping algebra, weobtain:

π(Aσ0

i1. . . Aσ0

in

)= π(Aσ0

i1) . . . π(Aσ0

in)

and since π is bijective, the Aσi i generate Jσ and hence their products generate the envelopingalgebra, which implies that π is surjective.Note that π cannot be trivial, hence if the universal enveloping algebra is simple, π must beinjective, as it is a representation of a simple associative algebra, which must necessarily beirreducible. In particular, the enveloping algebra of any embedding of a Jordan algebra withsimple universal envelope must be simple.

Note that in passing, this result also proves the intuition that the universal embedding has thehighest dimensional envelope for an embedding, since otherwise the extension from the universalembedding to the envelopes cannot be surjective.

31


Corollary 3.13. In the setting of lemma 3.12, we have in particular: Given two unital Jordanalgebras Ji, i = 1, 2 of self-adjoint matrices such that the universal algebra is simple, π extendsto a *-isomorphism of the corresponding enveloping algebras.

Proof. Given two unital Jordan algebras of self-adjoint matrices, the enveloping algebras areisomorphic by a *-isomorphism, which extends the Jordan-*-isomorphism between the two Jordanalgebras. To see this, call the two Jordan algebras J1 and J2. Note that since the Jordan algebrasare isomorphic, they necessarily have the same universal algebraU and universal embeddingsJσ1 and Jσ2 . Then, concatenating the two isomorphisms from the universal envelope gives therequired result. For clarity, look at the following diagram:

J S(J , σ)

S(J , σ0)

∃ h hom.

∃ g hom.σ0

σ

Here, the h is an extension of h via the universal property of S(J1, σ1) and H is an extension ofh via the universal property of S(J2, σ2). Since both are homomorphisms, they must be inversesof each other, thus proving the claim.

The fact that a representation of the Jordan algebra extends to a representation of the universalenvelope taken together with the fact that representations of simple algebras are always irreducible,we obtain the following corollary, which will be very important in the last section of this chapter:

Corollary 3.14. Given an embedded Jordan algebra J ⊂ Md, such thatMd is the universalenvelope and given another representation of J intoMd′ such that S(J) =Md′ , then d = d′. Inparticular, the two enveloping algebras are isomorphic as C*-algebras.

Proof. Denote the universal representation by σ0 and the second representation by σ. By thelemma, the Jordan-*-isomorphism h : Jσ0 → Jσ extends to a surjective representation of theenveloping algebras h : Md → Md′ . Now via surjectivity, d′ ≤ d. Suppose d′ < d, then hcannot be injective, hence is either trivial, which is impossible, since it extends h, which is anisomorphism, or has a nontrivial kernel, which is notMd. But this is impossible sinceMd has nonontrivial ideals and the kernel of an algebra homomorphism is an ideal. Hence d = d′.

This corollary is crucial, because we can now combine it with the well-known Skolem-Noethertheorem to classify certain classes of Jordan-representations. Let us recall that a complexassociative algebra is called central-simple, if it is simple and its centre (i.e. its set of commutingelements) is C1. We only state a weaker version adapted to our needs for reference (this versioncan be easily adapted from the version and proof in [30]):

32

Martin Idel Universal embeddings of finite dimensional Jordan algebras

Theorem 3.15 (Skolem-Noether theorem, [18]). Let A be a finite dimensional central-simpleassociative algebra over C. Every automorphism ofA is inner, i.e. every automorphism is of theformA 3 A 7→ S AS −1 where S ∈ A is an invertible element.

Corollary 3.16. Given the setting of the Skolem-Noether theorem, if the automorphism in questionis a *-automorphism, then S must necessarily be unitary.

Proof. If the automorphism is a *-automorphism, we have for every self-adjoint A ∈ A,(S AS −1)† = S −†AS † = S AS −1, hence S † = S −1.

Using this together with corollary 3.14 implies:

Corollary 3.17. Given a Jordan algebra Jσ0 ⊂ Md such that its enveloping algebra is Md.Then every other faithful representation of Jσ has either an enveloping algebra that is notcontained inMd or is equivalent to this representation in the sense that the correspondence isgiven by Jσ 7→ SJσ0S † with S ∈ Md a unitary matrix.

Proof. According to corollary 3.14, either we have a faithful representation with a bigger en-veloping algebra or else the enveloping algebras are isomorphic, where the isomorphism is givenas an extension of the corresponding Jordan-isomorphism aσ0 7→ aσ. ButMd is a central-simpleassociative algebra (since the only matrices that commute with all matrices are multiples of theidentity, cf. Schur’s lemma), hence the Skolem-Noether theorem implies the existence of S .

3.3. Universal embeddings of finite dimensional Jordanalgebras

We now need to study the universal enveloping algebras of simple Jordan algebras. However, wehave not yet given a classification of simple Jordan algebras, only of their self-adjoint parts (theformally real algebras in theorem 2.13). But this turns out sufficient:

Lemma 3.18. Given a finite-dimensional, simple Jordan algebra J and denote its self-adjointelements by Jh, then J = Jh + iJh, i.e. every element A ∈ J has a unique decomposition into aself-adjoint part and a skew-Hermitian part within the Jordan algebra.

Proof. We will show (see corollary (4.3) for the crucial uniqueness part) that for every Jordanalgebra J , there exists a unique trace-preserving, linear, positive projection T such that T (J) =

J . SinceMd is spanned by its self-adjoint elements, via linearity, T |(Md)h uniquely defines T .Any self-adjoint part of a finite-dimensional simple Jordan algebra J now defines a Jordanalgebra J ′ via J ′ := Jh + iJh. This is obviously a complex linear space and it is also a Jordanalgebra since:

(A + iB) ∗ (C + iD) = A ∗C − B ∗ D + i(B ∗C + A ∗ D) ∈ J ′ ∀(A + iB), (C + iD) ∈ J ′

But then, J ′ = J , since there is only one trace-preserving, linear, positive projection T such thatT ((Md)h) = Jh.

33


We have just seen that the finite-dimensional simple Jordan algebras are characterized by theirself-adjoint part, for which we have a structure theorem. Finally, we discuss the envelopingalgebras of these simple algebras (this part is a reproduction of [15] with adapted notation andsome small extensions. We include it here for completeness):We will make extensive use of the multiplication table of matrix units. Recall the standardbasis for complex matrices Ei ji, j=1,...,d, where Ei j is a matrix with a 1 at the (i, j)-th entry andzeros everywhere else. Then the following relations characterize these matrix units (for alli, j, k, l = 1, . . . d if not otherwise specified):

E2ii = Eii, EiiE j j = 0 ∀i , j, EiiE jk = δi jEik

E jkEii = δkiE ji, Ei jEkl = 0 for i, j, k, l distinct(3.10)

We will need these relations quite often:

Lemma 3.19 ([15], Case A). Let Jh = (Md)h for d ≥ 3, then J = Md and the universalembedding is given as A 7→ diag(A, AT ) with the universal enveloping algebra diag(A, B) withA, B ∈ Md, i.e. U(Md) =Md ⊕Md.

Proof. We follow the proof from [15]:The basic idea is the following: Knowing that a universal embedding exists (proposition 3.10)and knowing that it is the biggest possible envelope (remark after lemma 3.12), we will constructminimal conditions for the envelope of a Jordan algebra and then give an embedding such thatthe envelope has these conditions.To this end, consider the usual orthonormal basis ofMd(C), Ei ji, j=1,...,d, where only the (i, j)-thentry is nonzero and hence 1. Then we have:

2Ei j ∗ Ekl = δ jkEil + δilEk j (3.11)

Now we consider a representation of the Ei j intoMd′ for some d′ ∈ N, i.e. we consider Eσi j,

where the Eσi j fulfill relation (3.11) instead of the usual matrix relations in an associative algebra.

In particular, the Eσii are orthogonal idempotent elements in the algebra, meaning Eσ

ii ∗ Eσj j = 0 for

i , j and (Eσii )2 = Eσ

ii . Now we consider the universal enveloping algebra. Here, the Eσii remain

idempotents and since Eσii Eσ

j j = (Eσii )

2Eσj j = −Eσ

ii Eσj jE

σii = Eσ

j j(Eσii )

2 = Eσj jE

σii , the Eσ

ii are alsoorthogonal in the associative algebra. Now define for i , j

Gi j := Eσii Eσ

i jEσj j (3.12)

Then we obtain using (3.11):

Gi j = Eσii Eσ

i jEσj j = (Eσ

i j − Eσi jE

σii )Eσ

j j = Eσi jE

σj j (3.13)

Gi j = . . . = Eσii Eσ

i j (3.14)

34


Using Eσj jE

σkk = Eσ

j j ∗ Eσkk = 0 for j , k, we furthermore obtain Gi jGkl = 0 and if i, j, k are not

equal, then

Gi jG jk = Eσii Eσ

i jEσjkEσ

kk = Eσii (Eσ

ik − EσjkEσ

i j)Eσkk = Gik (3.15)

using that the Eii is an idempotent and Eii ∗ E jk = 0. Now define:

Gii := Gi jG ji

where we first have to prove that this is well defined, in the sense that this is independent of j.This is done by using eqn. (3.15) in a mere shifting of the argument (i , j , k):

Gi jG ji = Gi j(G jkGki) = (Gi jG jk)Gki = GikGki

Since Eσii Gi j = Gi j = Gi jEσ

j j, using Eσii Eσ

j j = Eσii ∗ Eσ

j j = 0 we obtain:

GiiG jk = 0 = Gk jGii (3.16)

if only i , j. Also, let i , j and choose k , i, j, then:

GiiGi j = (GikGki)Gi j = GikGk j = Gi j

Also, for i, j, k distinct, we obtain:

G2ii = Gi jG jiGikGki = Gi jG jkGki = Gii

G jkGii = δkiG ji

by making extensive use of eqn. (3.15). These relations form a complete set of matrix units (seeequations (3.10)) of the form like Ei j (not the representations Eσ

i j, which we know nothing about),i.e. the Gi j form the basis of a matrix algebraMd.Now we consider a second set of matrix units:

Gd+i,d+ j = EiiE jiE j j

Gd+i,d+i = Gd+i,d+ jGd+ j,d+i(3.17)

The crucial observation is now that eqn. (3.11) does not change by just changing the order of twosubscripts, hence we can do exactly the same calculations as above to obtain that the set in eqn.(3.17) is also a set of matrix units.We will now prove that Gi jGd+k,d+l = Gd+k,d+lGi j = 0 for all i, j, k, l = 1, . . . , d, hence the twocomplete sets of matrix units form a direct sum of two full matrix algebras. To see this, note thatfor i, j, k distinct Eσ

i jEσkk = 0 = Eσ

kkEσi j in the same way as we have proven that the orthogonal

idempotents E j j are also orthogonal in the associative envelope. But then, using eqn. (3.14):

Gi jGd+ j,d+k = Eσi jE

σj jE

σk jE

σkk = Eσ

i jEσk jE

σkk = −Eσ

k jEσi jE

σkk = 0

35


where in the second to last step we used (3.11). This implies also (i, j, k distinct)

Gi jGd+ j,d+i = Gi jGd+ j,d+kGd+k,d+i = 0

Gi jGd+ j,d+ j = Gi jGd+ j,d+iGd+i,d+ j = 0

Furthermore, since GiiEσii = Gii and Eσ

j jGd+ j,d+ j = Gd+ j,d+ j again via eqn. (3.14) and the analoguefor the second set of matrix units, we have GiiGd+ j,d+ j = 0 for i , j (the argument is the same asfor equation (3.16)).Putting everything together, using GiiGi+d,i+d = Gi j(G jiGd+i,d+k)Gd+k,d+i = 0, we have therequired Gi jGd+k,d+l for all i, j, k, l. Since the arguments are the same for Gd+k,d+lGi j, we havetwo orthogonal sets of matrix units.We can now define G :=

∑i Gii and G′ :=

∑i Gd+i,d+i, which are identities for Gi j and Gd+i,d+ j

respectively and we have just seen that it holds:

GG′ =∑

i j

GiiGd+ j,d+ j = 0 (3.18)

Now we denote by A(i) the subspace of the envelopeU of the Jordan algebra, generated by theGi j for i = 1 and the Gd+i,d+ j for i = 2. Then the A(i) are full orthogonal matrix subalgebras,since:

A(1)A(2) = 0 = A(2)A(1) (3.19)

via GG′ = G′G = 0, hence A(1) ⊕ A(2) ⊂ U. Now

Eσi j = Eσ

ii Eσi j + Eσ

i jEσii = Gi j + Gd+ j,d+i (3.20)

Eσii = . . . = Gii + Gd+i,d+i (3.21)

hence the Ei j that span J are elements of A(1) ⊕ A(2) or in other words, J ⊂ U ⊂ A(1) ⊕ A(2),sinceU is the smallest associative algebra containing J . This impliesU = A(1) ⊕ A(2). Hencethe universal envelope is a direct sum of two algebras, each of which is either 0 or isomorphic toMd, since it is a simple matrix algebra.We now need to construct the universal embedding. If we find an embedding, such that theenvelope isMd ⊕Md, we are finished with the proof, since the universal envelope has maximaldimension. To this end consider:

Eσi j :=

(S i j 00 S ji

)where the S i j are the canonical d × d-basis ofMd(C) (above we called this basis of canonicalmatrix units Ei j, however here this symbol has already been used). With this choice, sinceS i jS kl = δ jkS il, we can see that:

Gi j =

(S i j 00 0

)

36


Gd+i,d+ j =

(0 00 S ji

)Furthermore, the Jordan algebra is given by the elements diag(A, AT ), where A ∈ Md(C). Hencewe have a universal embedding.

The case d = 1 is the same as the the case d = 1 for symmetric matrices, since the self-adjointcomplex numbers are the real numbers and the case d = 2 corresponds, as we have already seen,to a spin factor. We had to exclude it here, since at one point we needed i, j, k all distinct, whichis impossible for d = 2. We will cover this case together with the spin factors, but it seems alsopossible to just give a similar but shorter proof for the special case of d = 2 dimensions.

Lemma 3.20 ([15], Case B). Let d ∈ N, Jh = (Md(R))h + i(Md(R))h (i.e. the symmetricmatrices), then the universal embedding is the canonical inclusion and the universal envelopingalgebra isMd.

Proof. We follow the proof from [15]:The basic idea is again: Knowing that a universal embedding exists and knowing that it is thebiggest possible envelope, we will construct minimal conditions for the envelope of a Jordanalgebra and then give an embedding such that the envelope has these conditions.Denote by E jk j,k=1,...,d the usual orthonormal basis ofMd(C). Then defining

F jk :=12

(E jk + Ek j) j, k = 1, . . . , d

we obtain a basis F jk j,k of J . Then by using Ei jEkl = Eilδ jk, we obtain:

F2ii = Fii 4F2

i j = Fii + F j j i , j

2Fii ∗ Fi j = Fi j i , j 4Fi j ∗ Fik = F jk i, j, k distinct

Fii ∗ F j j = Fii ∗ F jk = Fi j ∗ Fkl = 0 i, j, k, l distinct

These relations are the defining relations for the basis of J .We organize them in the equation:

4Fi j ∗ Fkl = δ jkFil + δ jlFik + δikF jl + δilF jk (3.22)

In particular, the first relation shows that Fii are idempotents that are orthogonal by the lastrelation.Now we consider an arbitrary embedding Fσ

i j and want to construct the enveloping algebraU.Again, the Fσ

ii must be idempotents of U, since they are idempotents of the Jordan algebra.Furthermore, Fσ

ii Fσj j = −Fσ

j jFσii . Also, Fσ

ii Fσj j = (Fσ

ii )2Fσj j = −Fσ

ii Fσj jF

σii = Fσ

j j(Fσii )2 = Fσ

j jFσii ,

hence Fσii Fσ

j j = 0 and the idempotents must also be orthogonal in the associative algebra. Nowdefine:

Gii := Fσii ; Gi j := 2Fσ

ii Fσi jF

σj j; ∀i, j = 1, . . . d (3.23)

37


Then G2ii = Gii, GiiG j j = 0 since we have orthogonal idempotents. Furthermore (for all i, j, k),

making use of (3.22):

GiiG jk = 2Fσii Fσ

j jFσjkFσ

kk = δi jG jk

G jkGii = δikG ji

where we use that the Fii are orthogonal idempotents. Putting the two equations together, weobtain also Gi jGkl = 0 if j , k. If i , j, we have:

Gi j = 2Fσii Fσ

i jFσj j = 2(Fσ

i j − Fσi jF

σii )Fσ

j j = 2Fσi jF

σj j

as well as Gi j = 2Fσii Fσ

i j. Then, making use of (3.22)

Gi jG ji = 4Fσii (Fσ

i j)2Fσ

ii = Fσii (Fσ

ii + Fσj j)F

σii = Fσ

ii = Gii

and for i, j, k distinct, we have

Gi jG jk = 4Fσii Fσ

i jFσjkFσ

kk = 2Fσii (Fσ

ik − FσjkFσ

i j)Fσkk = 2Fσ

ii FσikFσ

kk = Gik

But then the Gi j, for i, j = 1, . . . , d form a set of matrix units (see equations (3.10)). Denote byAthe algebra defined by the Gi j. Then, naturally,A ⊂ U by construction. On the other hand, wehave

Fσi j = 2(Fσ

ii ∗ Fσi j) = Fσ

ii Fσi j + Fσ

i jFσii =

12

(Gi j + G ji)

which implies U ⊂ A and in consequence U = A. In particular, the envelope is either 0 orMd sinceA is a simple algebra. Since it cannot be trivial, the universal embedding must haveenvelopeMd.To give another argument, if we take the extension to a complex Jordan embedding of thenatural embedding of the symmetric matrices (Md(R))h → (Md(C))h, then the envelope isMd

since the real symmetric matrices generateMd(R), hence the complex Jordan algebra generatesMd(C).

Lemma 3.21 ([15], Case C). Let d ≥ 3 and Jh =Md(H) and consider its embeddings, then theuniversal algebra of J isM2d and the universal embedding is given in proposition 3.6.

Proof. We follow again [15].The strategy is again the same as in the two previous proofs. We know that a universal embeddingexists and it is the embedding with the envelope of highest dimension, so we take an arbitraryembedding and search for constraints on the envelope such that in the end we have a set ofpossible envelopes. Then, we only need to find an embedding for the biggest envelope.We have already constructed an embedding of quaternion Hermitian matrices in proposition 3.6.

38


There, we have also seen a characterization of the complex Jordan algebra generated by thisembedding in eqn. (3.5) or eqn. (3.4). We recall that we have for an A ∈ M2d such that A ∈ J :

A =

(A11 A12A21 A22

)and the conditions:

A11 = AT22, AT

12 = −A12, AT21 = −A21

Hence, if the Ei j are the usual matrix units, we can span J by the following basis:

Hi j := Ei j + Ed+ j,d+i

Fi j = −F ji := Ei,d+ j + E j,d+i

Di j = −D ji := Ed+i, j − Ed+ j,i i, j = 1, . . . , d

The first observation is that the Hi j generate a subalgebra of J that is isomorphic toMd (whichis clear since they were constructed from the embedding in proposition 3.6). Also, we easilyderive the following multiplication table:

2Hi j ∗ Hkl = δ jkHil + δilH jk

2Hi j ∗ Fkl = δ jkFil + δ jlFki

2Hi j ∗ Dkl = δilDk j + δikD jl

2Fi j ∗ Dkl = δ jkHil + δilH jk − δikH jl − δ jlHik

2Fi j ∗ Fkl = 2Di j ∗ Dkl = 0

(3.24)

Now we consider the Hi j, Fi j,Di j as defining relations for J and we want to study an arbitraryrepresentation σ, i.e. we study Hσ

i j, Fσi j,D

σi j.

Let us define, similarly to the cases above and choosing i , j whenever they appear together:

Gi j := Hσii Hσ

i jHσj j Gii := Gi jG ji

Gd+i,d+ j := Hσii Hσ

jiHσj j Gd+i,d+i := Gd+i,d+ jGd+ j,d+i

Gi,d+ j := Hσii Fσ

i jHσj j Gi,d+i := GiiG j,d+i

Gd+i, j := Hσii Dσ

i jHσj j Gd+i,i := Gd+i, jG ji

and for all i, j = 1, . . . d. The idea is that the Hσi j form an embedding ofMd(C) (if we look at

proposition 3.6 the blocks on the diagonal form a universal embedding ofMd(C)), hence the firsttwo lines mimic the construction in the proof of lemma 3.19, while the last two lines provide theoff-diagonal elements, thus obtaining a full matrix algebra.To complete the proof, we now need to see that (denoting byA the associative algebra spannedby the G)

39


(a) these form a set of matrix units ofM2d (A ⊂ U)

(b) the elements Hσ, Fσ,Dσ that span J lie inside the space spanned by the G (U ⊂ A)

Part (a) is already established for Gi j and Gd+i,d+ j in the proof of Case A. This means we have forall i, j, k, l:

Gi jGkl = δ jkGil

Gd+i,d+ jGd+k,d+l = δ jkGd+i,d+l

Gi jGd+k,d+l = Gd+i,d+ jGkl = 0

(3.25)

We also recall Gi j = Hσii Hσ

i j = Hσi jH

σj j and the same for Gi+d, j+d. For i , j, we have (using

equations (3.24)):

Gi,d+ j = Hσii Fσ

i jHσj j = (Fσ

i j − Fσi jH

σii )Hσ

j j = Fσi jH

σj j

since the Hσii Hσ

j j = 0 for i , j as the Hσii are orthogonal idempotent elements. Similarly

Gi,d+ j = Hσii Fσ

i j and Gd+i, j = Dσi jH

σj j. For i, j, k distinct (recall d ≥ 3), we have:

Gk jG j,d+i = HσkkHσ

k jFσjiH

σii = Hσ

kk(Fσki − Fσ

jiHσk j)H

σii = Hσ

kkFσkiH

σii = Gk,d+i

i.e. Gi jG j,m+i = Gi jG j,kGk,d+i = GikGk,d+i for all j Likewise G j jG j,d+i = G j,d+i since

G j jG j,d+i = G jkGk jG j,d+i = G jkGk,d+i = G j,d+i

and these relations imply (using equation (3.25)) for all h, k, i, j provided that i , j:

GhkG j,d+i = GhkG j jG j,m+i = δ jkGh jG j,d+i = δ jkGh,d+i

⇒ GhkGi,d+i = δikGh,d+i

In total, we have for all i, j, k, l:

Gi jGk,d+l = δ jkGi,d+l (3.26)

Furthermore, for i , j, using equation (3.24):

Gi,d+ jGd+ j,d+i = Hσii Fσ

i jHσi jH

σii

= −Hσii Hσ

i jFσi jH

σii

= Hσii Hσ

i jFσjiH

σii

= Gi jG j,d+i

= Gi,d+i

40


and similarly as above (same calculations), we establish for all i, j, k, l:

G j,d+iGd+h,d+k = δikG j,d+k (3.27)

Similar calculations lead to:

Gd+i, jGhk = δ jhGd+i,k

Gd+h,d+kGd+i, j = δikGd+h, j(3.28)

Also, using the orthogonality of Gi j and Gd+i,d+ j in equation (3.25), we obtain:

G j,d+iGhk = (G j,d+lGd+l,d+i)Ghk = G j,d+l(Gd+l,d+iGhk) = 0

for all h, i, j, k. Similarly, we obtain:

G j+d+iGhk = Gd+ j,iGd+h,d+k = G j,iGd+h,k = Gd+ j,d+iGh,d+k = 0 (3.29)

Furthermore, if i , k (again using the orthogonality relations):

Gh,d+kGd+i, j = Gh,d+kGd+k,d+kGd+i,d+iGd+i, j = 0

Then, for h, i, j distinct, we have (using (3.24)):

Gh,d+iGd+i, j = HσhhFσ

hiDσi jH

σj j

= Hσhh(Hσ

h j − Dσi jF

σhi)H

σj j

= HσhhHσ

h jHσj j + Hσ

hhDσi jH

σj jF

σhi

= HσhhHσ

h jHσj j − Dσ

i jHσhhHσ

j jFσhi

= Gh j

and

Gh,d+iGd+i,h = Gh,d+iGd+i, jG jh = Ghh

and so on as above to obtain for all i, j, k, l:

Gi,d+ jGd+k,l = δ jkGil

Gd+i, jGk,d+l = δ jkGd+i,d+l(3.30)

Now we put equations (3.26)-(3.30) together:

Gi jGkl = δ jkGil

Gi jGk,d+l = Gi,d+ jGd+k,d+l = δ jkGd+l

Gi jGd+k,d+l = Gd+k,d+lGi j = 0

41


Gi jGd+k,l = Gd+i,d+ jGl,d+k = Gi,d+ jGk,l = Gd+i, jGd+k,d+l = 0

Gd+i,d+ jGd+k,d+l = δ jkGd+i,d+l

Gd+i,d+ jGd+k,l = Gd+i, jGk,l = δ jkGd+i,l

Gd+i, jGk,d+l = δ jkGd+i,d+l

Gi,d+ jGd+ j,l = δ jkGil

and we see that the Gi j,Gd+i,d+ j form a complete set of 2d × 2d-matrix units. In particular, theassociative algebraA spanned by Gi j,Gi,d+ j,Gd+i, j,Gd+i,d+ j is simple is a simple C*-algebra ofdimension (2d)2 and fulfillsA ⊂ U for any envelopeU.For part (b) (U ⊂ A), we just rewrite the definitions:

Gi j + G j+d,i+d = Hσii Hσ

i j + Hσi jH

σii = Hσ

i j i , j

Gii + Gd+i,d+i = Gi jG ji + Gd+i,d+ jGd+ j,d+i = Hσii Hσ

i jHσjiH

σii + Hσ

ii HσjiH

σi jH

σii

= Hσii (Hσ

i jHσji + Hσ

jiHσi j)H

σii = Hσ

ii

Gi,d+ j −G j,d+i = Hσii Fσ

i j − FσjiH

σii = Hσ

ii Fσi j + Fσ

i jHσii = Fσ

i j

Gi+d, j −G j+d,i = Hσii Dσ

i j − DσjiH

σii = Hσ

ii Dσi j + Dσ

i jHσii = Dσ

i j

henceU = A.This proves that the universal envelope is either 0, which is impossible, orM2d, since it is asimple algebra with (2d)2 generators. This implies that the universal algebra is M2d and theembedding given in prop. 3.6 is universal and generatesM2d as the envelope.

We note that the case d = 1 corresponds again to the symmetric elements, since the self-adjointquaternions are the real numbers.The case d = 2 will turn out slightly different from the other cases, its universal embedding isdifferent, as we shall see from studying spin factors. The proof does not work here, because weused the proof of Case A here, to obtain that the Gi j and Gd+i,d+ j form two sets of matrix units.

Lemma 3.22 ([15], Case D). Let J be a spin factor of dimension d + 1. Then we have two cases:Either k = 2n is even, then U = M2n , or k = 2n + 1 is odd, then U = M2n ⊕M2n . Also, fork = 2, this is Case A for d = 3 and for k = 5 this is Case C for d = 2 that we have not yet covered.The universal embedding is always given by proposition 3.7.

Proof. The strategy is again the same as in the two previous proofs. We know that a universalembedding exists and that it is the embedding with the envelope of highest dimension, so we takean arbitrary embedding and search for constraints on the envelope such that in the end we have aset of possible envelopes. Then, we only need to find an embedding for the biggest envelope.We have already established a representation of spin factors in the first section of this chapter.Taking the basis of the spin-system to be S 1, . . . , S k and denoting S 0 := 1 as before, we have the

42


multiplication table:

S 0 ∗ S i = S i,

S 2i = S 0, i = 1, . . . , k

S i ∗ S j = 0, i, j = 1, . . . , k; i , j

(3.31)

We first consider the case k = 2n even.The universal algebraU is the algebra of Clifford numbers, where we claim that the basis is givenby

S ε11 . . . S εd

d

where ε j ∈ 0, 1 for all j. The relations in equation (3.31) let us reorder any product of S i intoproducts of the form S ε1

1 . . . S εdd , hence these must span the algebra. Now suppose the products

were not linearly independent. Then we have:

S ε11 . . . S εd

d =∑

(εi1 ,...,εid )

cεi1 ,...,εidSεi11 . . . S

εidd

with complex numbers cεi1 ,...,εidand some set of εi j . Without loss of generality, we assume that

the terms on the right hand side are all linearly independent and that the number of nonzero εi iseven. Now every S i anticommutes with S j for i , j. This means that every S i such that εi = 0commutes with S ε1

1 . . . S εdd and every S i such that εi = 1 anticommutes with S ε1

1 . . . S εdd , hence the

same must hold on the right hand side. In particular, it must hold for every summand individually.Pick one summand S

εi11 . . . S

εidd , then a quick calculation shows that S

εi11 . . . S

εidd = S ε1

1 . . . S εnd

necessarily. Hence the products above form a basis and the Clifford algebra has dimension 22m

by construction.Now introduce the following objects:

Um := S 1S 2 . . . S 2m−1, Vm := S 2m−1S 2m

The basis relations eqn. (3.31) implie:

U2m = (−1)m−1, V2

m = −1, UmVm = −VmUm

Call Qm the algebra generated by (S 0,Um,Vm,UmVm). It is easy to see that Qm is isomorphic toM2. The elements S 1, S 2, . . . , S 2m−2 commute with Um and Vm and generate a Clifford algebra ofdimension 22(m−1), which we want to callW. Furthermore, every basis element A = S ε1

1 . . . S εdd

fulfills: A ∈ Qm, A ∈ W or A = QW where Q ∈ Qm and W ∈ W. Hence we have a tensorproduct structureU = Qm ⊗W. Introducing again:

Uk := S 1, . . .U2k−1, Vk := S 2k−1S 2k

⇒ U2k = (−1)k, V2

k = −1

43


Qk := (S 0,Uk,Vk,UkVk)

an induction argument givesU = Q1 ⊗ Q2 ⊗ . . . ⊗ Qm.Observe that in Qk, the only elements commuting with all other elements are multiples of theidentity, hence also for U, the centre must be trivial. In particular, U is a central C*-algebraof dimension 22m, which by the Artin-Wedderburn structure theorem means that U M2m

(assumingU , 0, which is clear). In particularU is simple.Now let d be odd, i.e. d = 2m + 1.Defining C := S 1 . . . S 2m+1, we obtain that C commutes with every S i, hence C is in the centre ofU. We have C2 = (−1)m and by regarding iC for odd m instead of C, we can therefore assumeC2 = 1.The elements S 1, . . . S 2m generate a Clifford algebraW which is central simple of dimensionality22m using the case d even. Since CS 1 . . . S 2m ∝ S 2m+1, we have

WC(C) = C(C)W = U

(The inclusion ⊃ is given by the observation CS 1 . . . S 2m ∝ S 2m+1 and the inclusion ⊂ is givenby construction). SinceW has trivial centre, this immediately implies that the centre of U isC = span(S 0,C). But then U is a direct sum of two simple algebras isomorphic toW, againusing the Artin-Wedderburn structure theorem 3.23.It could in principle be that any embedding has only a simple envelope, as C could be the identity.However, we will show that the embedding given in proposition 3.7 has an envelope that is adirect sum of two full algebras. To see this, we follow [12] (paragraph 6.2.2):Note that the C*-algebra generated by σ3 and 1 is just the diagonal matrices. If we add σ1, wespanM2 since:

σ2 ·12

(1 + σ3) =

(0 01 0

)12

(1 + σ3) · σ2 =

(0 10 0

) (3.32)

hence the self-adjoint part of the spin factor spans the real part ofM2, i.e. 1, σ3, σ2 spansM2as complex algebra.This provides the start for an induction: If we know that for some positive integer m, S 1, . . . , S 2m

embedded via

S 1 := σ3 ⊗ 1m−1

S 2 := σ1 ⊗ 1m−1

...

S 2m−1 := σm−12 ⊗ σ3

S 2m := σm−12 ⊗ σ1

44

Martin Idel Representations of the different types of Jordan algebras

spanM2m , then considering the natural embedding σ : M2m → M2m+1 as for proposition 3.7,namely:

S σ1 := S 1 ⊗ 1 . . . S σ

2m := S σ2m ⊗ 1

by definition of S 2m+1, 1 and S 2m+1 span the diagonal 2 × 2 matrices with entries inM2m . Butthen, the embedding σ of the spin factor spansM2m ⊕M2m , which is what we claimed.If we then add S 2m+2, we again spanM2m+1 completely, since 1, S 2m+1, S 2m+2 span the 2 × 2matrices with entries inM2m by using (3.32).Hence by induction, for every k = 2n + 1, we can see that the envelope of the given embedding isM2n ⊕M2n , which implies that this must be the universal envelope as claimed.

Now we want to have a look at the cases we left out for the Hermitian matrices.First we have a look at k = 3. Then we can choose the basis:

S 0 = 1, S 1 = σ3, S 2 = σ1, S 3 = σ2

and the universal algebra is a direct sum of two algebras, each of which is isomorphic to theClifford algebra determined by S 1, S 2. But as observed in equation (3.32), this algebra is a full2 × 2-matrix algebra, hence V3 = M2(C). The universal algebra and embedding of M2 aretherefore given by the universal algebra and embedding ofV3, but this is the same result as forMd(C) and d , 2.Now consider an embedding of the case k = 5 via:

S 0 := 1, S 1 := diag(σ3, σ3)

S 2 := diag(σ2, σT2 ), S 3 := diag(σ1, σ1)

S 4 :=(

0 σ2σ2 0

), S 5 := i

(0 σ2−σ2 0

)Then a quick calculation shows that this is a complex extension of an embedding of (M2(H))h,hence the missing case in Case C. Since the S i fulfill the Clifford algebra (as another quickcalculation shows), the universal algebra of (M2(H))h is a direct sum of two 4 × 4 algebras. Thisis different from the universal embeddings of (Md(H))h for d , 2.

3.4. Representations of the different types of Jordan algebras

Having computed the universal envelopes for Jordan algebras, we now proceed to looking atthe embeddings of Jordan algebras inside their universal envelope. In order to get a completeclassification of Jordan algebras, we need all possible embeddings. We already know how theuniversal embeddings look like and that all universal embeddings are isomorphic by algebraautomorphism of the enveloping algebra. The universal envelopes are now embedded as C*-algebras into someMd. We will not develop the theory of embeddings of C*-algebras here, since

45


this lies outside the scope of this thesis; we only recall the usual Artin-Wedderburn structuretheorem:

Theorem 3.23 (Artin-Wedderburn structure theorem; [40] with proof in [39]). Given a unitalC*-subalgebraA ofMd(C), then this algebra has the form:

A =

K⊕i=1

Mdi ⊗ 1mi (3.33)

where the mi are called the multiplicities and the di are the dimension of the full matrix-subalgebras of A. In particular, any nontrivial, unital representation of a simple *-algebraA is isomorphic to

A =Md′ ⊗ 1m (3.34)

with d′m = d.

This theorem tells us that equation (3.2) of our basic structure theorem 3.3 holds, withoutspecification of the terms J . This will then be done in section 3.4, where we will need theArtin-Wedderburn theorem, again.The usual form of the Artin-Wedderburn theorem (c.f. [39]) is a little bit stronger in the sensethat it asserts that all embeddings of a C*-algebra are automorphic, however we do not care aboutgeneral algebra automorphisms, but only those that extend to an automorphism of the underlyingMd. More precisely:

Definition 3.24. We call two embeddings σ0 ,σ1 : J →Md of a Jordan algebra J equivalent, ifthere is an algebra automorphism S :Md →Md that restricts to a Jordan isomorphism on Jσ0 .

The discussion will be dominated by the Skolem-Noether theorem and we have to consider allpossible embeddings of Jordan algebras.

Lemma 3.25. Consider the universal embedding for the Jordan algebrasJ withJh = (Md(R))h

(given in lemma 3.20), Jh = (Md(H))h (given in lemma 3.21) and spin factors Jh = (Vk)h withk = 2n (given in lemma 3.22). Then any embedding is universal and all universal embeddingsare isomorphic up to isomorphism of the embedding of the envelope, i.e.:Given σJ → Md such that S(J , σ) =Md, then for any other embedding σ1 of the same type,we have Aσ1 = UAσU† for some U ∈ Md unitary.

Proof. The existence and form of the universal embedding follows from the cited lemmata. Sinceall embeddings must have either trivial envelope, which is impossible for an injective embedding,or the envelope of the universal embedding, since this is a simple matrix algebra, the lemmafollows directly from corollary 3.17.

46


With a bit more effort, we can classify the embeddings of complex Hermitian matrices. At firstsight this is quite surprising that this part is harder, since the embedding ofMd(C) as a Jordanalgebra seems to be obvious, but the universal algebra turned out to be more complicated inlemma 3.19:

Lemma 3.26. Let J =Md(C) and consider embeddings σ : J →Md′ with d′ ∈ N. There aretwo types of possible embeddings:

• d′ = dk for some positive integer k and σ is inclusion induced σ : (Md)h → (Md)h ⊗ 1k.

• d′ = dk + dl with k, l ∈ N and the embedding is given by σ : (Md)h → (Md′)h, where theembedding is given by

σ : (Md)h →Md ⊗Mk ⊕Md ⊗Ml

A 7→ A ⊗ 1k ⊕ AT ⊗ 1l(3.35)

The first case comprises all possible embeddings where the envelope is a full associative algebras,while the second case comprises all universal embeddings of the Jordan algebra.Given another embedding σ : (Md(C))h → (Md′)h, we can find a unitary U ∈ Md′ such thatUMd(C)σU† =Md(C)σ for one of the embeddings σ above.

Proof. We know that every faithful representation extends to a surjective homomorphism of theuniversal enveloping algebra to the enveloping algebra of the embedding. Via 3.19, this envelopeis a direct sum of two simple algebras. The kernel of the homomorphism must be a subalgebra,i.e. either 0, one of the two summands or the whole algebra. Since the representation is faithful,the last case is excluded.Consider first a nonuniversal embedding σ, i.e. the kernel of the homomorphism of the universalenveloping algebra to the enveloping algebra of the embedding σ is nontrivial. Then, since theuniversal envelope is a direct sum of two simple algebras, the envelope S(Md(C), σ) must beMd via corollary 3.14. But thenMd(C)σ =Md, i.e. the Jordan algebra is a C*-algebra and allembeddings are given by the Artin-Wedderburn theorem. Via lemma 3.17, we have that these areall the possible representations up to isomorphism of the enveloping C*-algebra.In the case where the kernel of the homomorphism is trivial, the correspondence aσ0 7→ aσ for alla ∈ J with the universal embedding σ0 and some other embedding σ extends to an isomorphismon the level of enveloping algebras.Now we consider the embedding given by

σ0 : (Md)h →Md ⊗Mk ⊕Md ⊗Ml

A 7→ A ⊗ 1k ⊕ AT ⊗ 1l

This embedding is universal by lemma 3.19. A universal embedding intoMd′ now consists of aconcatenation of an embedding into its envelope (call it σ1 : (Md(C))h → S((Md(C))h, σ1)) and

47


an embedding of the envelope intoMd′ (call it σ2 : S((Md(C))h, σ1)→Md′(C)), i.e. any otherembedding σ is of the form σ = σ2 σ1.First, we know that σ1 is a universal embedding and any two universal embeddings are automor-phic. Furthermore, the envelope is a C*-algebra, hence its embeddings are given by the Artin-Wedderburn theorem 3.23. In principle, two C*-algebras of the same dimension are automorphic,which proves that any two universal embeddings of Jordan algebras are isomorphic, however theautomorphism need not extend to an automorphism ofMd′ (hence an inner-automorphism bySkolem-Noether theorem 3.15), i.e. the embeddings need not be equivalent. However, we want toclassify equivalent embeddings.Let d′ = dk + dl as in the lemma, then the Artin-Wedderburn theorem 3.23 tells us all embeddingsσ2 of the universal envelope. These are given by:

σ2 :Md ⊕Md →Md′

A ⊕ B 7→ U diag(A ⊗ 1k, B ⊗ 1l)U†

with U ∈ Md′ and embeddings of different k, l are inequivalent.Now consider the embeddings σ1 into the universal envelope. By definition, the correspondenceA⊕ AT 7→ Aσ1 for all A ∈ J extends to an automorphism h ofMd ⊕Md →Md ⊕Md. Since theautomorphism must send simple summands to simple summands and automorphisms of simplesummands are inner by the Skolem-Noether theorem 3.15, we obtain:

h :Md ⊕Md →Md ⊕Md

(A1 ⊕ A2) 7→2⊕

i=1

UiAπ(i)U†

i

where π is a permutation of 1, 2 and Ui ∈ Md is a unitary for i ∈ 1, 2. These are all possibleembeddings of J into S(J).First, if we consider embeddings σ1 with π = 1, then h extends to an automorphism ofMd′ , ifwe concatenate the embedding σ1 with an embedding σ2 of the envelope intoMd′ . In particular,there exist U ∈ Md′ , U1,U2 ∈ Md unitaries such that

σ2σ1 : J →Md′

A 7→U diag(U1AU†1 ⊗ 1k, (U2AU†2)T ⊗ 1l)U†

= U diag((U1 ⊗ 1k)(A ⊗ 1k)(U1 ⊗ 1k)†, (UT2 ⊗ 1l)(AT ⊗ 1l)(UT

2 ⊗ 1l)†)U†

= U(U1 ⊗ 1k ⊕ UT2 ⊗ 1l) diag(A ⊗ 1k, AT ⊗ 1l)(U1 ⊗ 1k ⊕ UT

2 ⊗ 1l)†U†

=: U diag(A ⊗ 1k, AT ⊗ 1l)U†

and U is also a unitary.Consider now the embeddings σ1 with π = (1, 2), then via a similar calculation, all embeddingsare given by

σ2σ1 : J →Md′

48


A 7→U diag(AT ⊗ 1k, A ⊗ 1l)U†

for some unitary U ∈ Md′ . But if we define U :=(

0 1dk

1dl 0

), then we obtain

U diag(A ⊗ 1k, AT ⊗ 1l)U† = diag(AT ⊗ 1l, A ⊗ 1k)

hence the embeddings were already covered by the case π = 1. But this proves that all possibleembeddings with an envelope that is a sum of two simple envelopes are given by equation(3.35).

Again, the case d = 1 is covered by the corresponding symmetric case.The last case to consider are the spin factors with k = 2n + 1. This is similarly difficult as thecomplex matrices, since the envelope is not simple. This implies that there will be inequivalentrepresentations, as in the case of complex matrices. We need this case to cover the case of d = 2quaternionic Hermitian matrices as well.Recall the universal embedding given in proposition 3.7. We will also give another embedding,which is just the universal embedding concatenated with an embedding of the envelope. Let k, lbe positive integers, let k = 2m + 1 for some m ∈ N and define the spin-system S 1, . . . , S 2m+1:

S 1 := σ3 ⊗ 1(m−1) ⊗ diag(1k,1l)

S 2 := σ1 ⊗ 1(m−1) ⊗ diag(1k,1l)

...

S 2m := σ(m−1)2 ⊗ σ1 ⊗ diag(1k,1l)

S 2m+1 := σ(m)2 ⊗ diag(1k,−1l)

Then we have the following lemma:

Lemma 3.27. ForVk a spin factor of dimension k + 1 where k = 2m + 1 and the spin factor isdefined by a spin-system S 1, . . . , S k. Given an embedding σ : Vk →Md for some d ∈ N, thenwe have two cases:

• d = 2mk + 2ml with positive integers k, l as above.

σ : Vk →Md

S i 7→ US iU†(3.36)

for some unitary U ∈ Md.

• d = 2mk, S 1, . . . , S k−1 form an embedding ofVk−1 intoMd and

S sn+1 = εS 1 . . . S 2m (3.37)

where ε = 1, i such that S 2n+1 is self-adjoint.

49


In particular, this covers also the case of 2 × 2 Hermitian matrices (which corresponds to thespin factorV3) and the case of the 2 × 2 quaternionic matrices (which corresponds to the spinfactorV5).

Proof. The fact that the universal algebra is as stated is Case D of the previous section (lemma3.22).Now proceed as for the case ofMd(C). Consider an arbitrary faithful representation σ ofVk. Aswe know that the Jordan isomorphism between the universal embedding σ0 and the embedding σ

extends to a homomorphism between the envelopes, the envelope ofVσk can be either 0, which

is impossible, since the representation is faithful,M2m ⊕M2m , or S(Vσk ) =M2m .

Consider first the case where the embedding is universal. Then we obtain equation (3.36) inexactly the same way as we obtained (3.35), the proof covers both cases.Finally, consider the case where the embedding has a simple envelope. The embedding is nowfully determined by the embedding of S σ

i for i = 1, . . . 2m + 1. Clearly, S 1, . . . , S 2m forms anembedding of the spin factorVk−1. Since we know that there is only one possible embedding ofthis type, it must be a universal embedding with simple envelopeM2m . Now consider

C := S 1S 2 . . . S 2m+1

Then, as in the proof of the universal envelope, C commutes with all S i for i = 1, . . . , 2n + 1,since C is a product of an odd number of S i. Hence C commutes with every element in theenvelope of Vσ

k , M2m−1 and C ∈ C(S(Vk), σ), but since this is a simple algebra, C = λ1 withλ ∈ C. Furthermore, using the anticommutation relations, we obtain:

C2 = (S 1 . . . S 2m+1)2 = (−1)2mS 2 . . . S 2m+1S 2 . . . S 2m+1

= (−1)∑2m

i=1 i1 = (−1)m(2m+1)

1

Hence we have two cases (via S 2i = 1 for all i, we know that the S i are invertible):

C2 =

1, m = 2n−1, m = 2n + 1

(3.38)

⇒ S 2m+1 =

±S 1 . . . S 2m, m = 2n±iS 1 . . . S 2m, m = 2n + 1

(3.39)

Hence we have constructed two embeddings (depending on the choice of sign for S 2m+1) for eacheven-dimensional spin factor. We need to see that these embeddings are isomorphic and that allother embeddings are also isomorphic to this embedding.Consider first two representations σ1 and σ2 such that S 2m+1 has the positive sign equation (3.39).Since S σ1

1 , . . . S σ12m as well as S σ2

1 , . . . S σ22m generate a spin factor Vk−1, we know that there

is a unitary U ∈ M2m such that US σ1i U† = S σ2

j for all i = 1, . . . , 2m and some j. But thenalso US 2m+1U† = ±S 2m+1 after reordering. Thus, every two representations are isomorphic. Inparticular, renumbering the different S i gives an isomorphism of the representation with a positivesign and a negative sign.

50


Example: To give a better intuition about what happens, we will give two embeddings as inproposition 3.8 with different sign in equation (3.39) and give their isomorphism:

σ1 : S σ11 = σ3 ⊗ 1

m−1 σ2 : S σ21 = S σ1

1

S σ12 = σ1 ⊗ 1

m−1 S σ22 = S σ1

2...

...

S σ12m−2 = σm−2

2 ⊗ σ1 ⊗ 1 S σ22m−2 = S σ1

2m−2

S σ12m−1 = σm−1

2 ⊗ σ3 S σ22m−1 = S σ1

2m

S σ12m = σm−1

2 ⊗ σ1 S σ22m = S σ1

2m−1

S σ12m+1 = εS σ1

1 . . . S σ12m S σ2

2m+1 = −εS σ21 . . . S σ2

2m

where ε = 1 for m even and ε = i for m odd.

Now consider U = 1√21

(m−1) ⊗

(1 11 −1

). We observe:

12

(1 11 −1

)σ3

(1 11 −1

)= σ1

12

(1 11 −1

)σ1

(1 11 −1

)= σ3

12

(1 11 −1

)σ2

(1 11 −1

)= −σ2

which implies:

US σ1i U† = S σ2

i ∀i = 1, . . . 2m

US σ12m+1U† = εUS σ1

1 . . . S σ12mU†

= εS σ21 . . . S σ2

2mS σ22m−1 = −εS σ1

1 . . . S σ22m

= S σ22m+1

which means that the two embeddings are isomorphic.This finishes the proof of the characterization of Jordan algebras as stated in theorem 3.3.Let us note the following corollary:

Corollary 3.28. Consider every embedding of Vk with odd k with a simple envelope (which wehave just characterized and which are given by proposition 3.8). Then for k ≤ 5, the embeddingsare reversible.

Proof. Denote the embedding by σ. Given the spin-system S 1, . . . , S k defining the spin factor(we drop the σ for readability), we need to see that

S i1 . . . S in + S in . . . S i1 ∈ Vσk ∀i1, . . . , in ∈ 1, . . . , k

51


Using the anticommutation relations, this amounts to showing that

S ε11 . . . S εk

k + S εkk . . . S ε1

1 ∈ Vσk ∀ε1, . . . , εk ∈ 0, 1

For k ≤ 3, these are just products of three generators, hence this follows immediately. For k ≤ 5,using that S 5 = ±S 1 . . . S 4, we can express S k in terms of the other S i, hence we need to see that

S ε11 . . . S ε4

4 + S ε44 . . . S ε1

1 ∈ Vσ5 ∀ε1, . . . , ε4 ∈ 0, 1

But then every product except S 1S 2S 3S 4 + S 4S 3S 2S 1 consists only of three or less generators,which implies that it lies in Vσ

k . Moreover S 1S 2S 3S 4 + S 4S 3S 2S 1 = ±2S 5 ∈ V5 since S 5 isself-adjoint. But then, Vσ

5 is reversible.

Finally, we need a proof of proposition 3.4:

Proof. First note that two equivalent embeddings are either both reversible or both irreversible.This means we only have to consider the different equivalence classes of embeddings for a giventype of Jordan algebra.LetJ =Md(R) for d ∈ N. Then the universal embedding is inclusion induced and since elementsof the form A1 . . . An + An . . . A1 for A1, . . . An ∈ J are obviously symmetric, J is reversible.The same argumentation holds for J =Md(C) for d ≥ 3 (the case of d = 2 being simultaneouslya spin factor, hence it will be done later on) and J =Md(H) for d ≥ 3 (again, the case d = 3 willbe done later on).For the spin factors, we basically follow the proof presented in [12]. A different proof can also befound in [10].From the Jordan product in spin factors, we can see that given the self-adjoint part of a spin factor(Vn−1)h and a basis S i ⊂ R

n−1, we have S i ∗ S j = δi j, meaning that the basis fulfills the Cliffordalgebra relation. By multilinearity, (ir)reversibility can be seen by considering just a basis of thespin factor.Now consider first the case n = 3, 4, i.e. the spin factor is generated by 1, S 1, S 2 and1, S 1, S 2, S 3. By the Clifford relations, any multiplication of some of these elements canbe brought to the form S ε1

1 S ε22 for n = 3 and S ε1

1 S ε22 S ε

3 for n = 4, where εi ∈ 0, 1 for alli = 1, 2, 3 for dimension n = 3, 4. In particular, for n = 3, 4, given

B1 . . . Bm + Bm . . . B1 ∈ (Vn−1)h

we can choose Bi = E j for some j ∈ 1, 2, 3 and all i ∈ 1, . . . ,m and we can then bring this intothe form E1 · E2 · E3 + E3 · E2 · E1 (for n = 4, for n = 3 the expression does not contain E3). ButE1 · E2 · E3 + E3 · E2 · E1 ∈ (Vn−1)h always, hence the two spin factors must be reversible.Now consider and embedding of (V4)h and assume that it was reversible. Consider againa basis 1, S 1, S 2, S 3, S 4 and define S 5 = S 1S 2S 3S 4. By the Clifford relations, we obtainS 5 = 1

2 (S 1S 2S 3S 4 + S 4S 3S 2S 1). However, we also obtain S 25 = 1 and S 5 ∗ S i = 0 always, hence

1, S 1, . . . , S 5 is a linear independent set in (V4)h which is obviously impossible.

52


Consider finally n ≥ 7. We assume again reversibility of the Jordan factor. As we have seen in thecase n = 5, given 1, S 1, S 2, S 3, S 4, assuming reversibilty of the Jordan factor, S 5 := S 1S 2S 3S 4must be contained in the algebra and fulfills the Clifford relations, i.e. can be taken as a basisvector. Now we add a sixth S 6 such that the Clifford relations are fulfilled, e.g. S 6 ∗ S i = 0 for alli ∈ 1, . . . 5. In particular:

S 5S 6 = −S 6S 5 = −S 6(S 1S 2S 3S 4) = −(S 1S 2S 3S 4)S 6 = −S 5S 6

but then already S 5S 6 = 0 which is impossible, since both S 5, S 6 are invertible elements. Henceany spin factor must be irreversible.As a last case, consider the case (V5)h. The universal embedding has been given in proposition3.7 and we have seen during the proof of theorem 3.3 that all embeddings with universal envelopeare isomorphic. Hence, any embedding with envelope that consists of a direct sum of two simplealgebras is isomorphic to the universal embedding.We consider first the embedding σ : V5 →M4 via:

S σ0 := 1, S σ

1 := diag(σ3, σ3)

S σ2 := diag(σ2, σ

T2 ), S σ

3 := diag(σ1, σ1)

S σ4 :=

(0 σ2σ2 0

), S σ

5 := i(

0 σ2−σ2 0

)which we have already encountered in chapter 3. Since the enveloping algebra must be isomorphicto eitherM4 orM4 ⊕M4, in this case, it must beM4. Moreover, we have seen in the proofof lemma 3.22 that this is an embedding ofM2(H) as in 3.6, which is reversible by a simplecalculation. Hence, σ must be reversible. We note that in particular S σ

5 := 12 (S 1S 2S 3S 4 +

S 4S 3S 2S 1)σ since:(σ3 00 σ3

)·

(σ2 00 σT

2

)·

(σ1 00 σ1

)·

(0 σ2σ2 0

)=

(0 iσ2−iσ2 0

)and the same for the other combination, hence the stated result. Now, consider another embeddingσ with S σ

i = S σi for all i = 0, . . . , 4 and

S σ5 := i

(0 −σ2σ2 0

)which is another embedding of V5, which is, by itself, also reversible and isomorphic to σ, sinceall embeddings with the same envelope are isomorphic. However, now consider the representation

σ0 : V5 →M8

A 7→(Aσ 00 Aσ

)(3.40)

53


then this embedding satisfies (using S 25 = 1 and S σ

5 = −S σ5 ):

12

(S 2S 2S 3S 4S 5 + S 5S 4S 3S 2S 1)σ0 =

(1 00 −1

)< Vσ0

5

hence this embedding is not reversible. Since its envelope is isomorphic to the universal envelope,this must the the universal embedding, which concludes the proof.

Final remarksWe have seen that every finite dimensional Jordan-*-algebra with an identity is a direct sum ofof Jordan algebras of six different types. Three of them result from a well-known embedding ofHermitian matrices over an associative division ring into the complex matrices, one of anotherembedding of Hermitian matrices over C intoMd(C) and these embeddings are reversible andunique. In addition, we have embeddings of spin factors, where we have one embedding for odddimensional spin factors and two embeddings for even dimensional ones. In higher dimension(≥ 7), these are always irreversible, while in lower dimensions, they cannot be. In fact, allreversible embeddings correspond to embeddings of Hermitian matrices, i.e. in order to obtainuniqueness of the decomposition, we had to exclude certain dimensions of Hermitian matrices(1 × 1 complex and quaternionic Hermitian and 2 × 2-complex and quaternionic Hermitian andspin factors of dimension ≤ 2), where an embedding can be seen as both, an embedding of a spinfactor or of some different types of Hermitian matrices. However, this is only a phenomenon inlower dimensions, where there is not enough space to accommodate all different types of Jordanalgebras.

54

4. Positive Projections onto fixed-pointalgebras

In this chapter, we want to analyze positive projections T ∗∞ onto unital Jordan algebras (from nowon, we will silently assume all Jordan algebras to be unital). These projections arise for examplefrom linear, positive and trace-preserving maps T with a full-rank fixed point. Let us first statethe result of this section and give an outline of the proof, which will cover the rest of the chapter.The reversible part of this theorem was basically proven in [8, 36], while parts of the irreversiblepart are proven in [10]. However, using theorem 3.3, we are able to give all possible projections,thereby including original results especially on spin factors.

Theorem 4.1. Let J =⊕K

i=1Ji ⊂ Md be a Jordan algebra with Ji ⊂ Mdi for i = 1, . . . ,K and∑Ki=1 di = d.

1. There exists a unique trace-preserving projection onto J .

2. Consider a positive projection onto a Jordan algebra J . Then every projection T :Md →

Md onto J is a sum of projections Ti :Mdi →Mdi onto Ji and we have two cases:

• Let Ji be simple with simple envelope S(Ji), then we have

Mdi

T (1)i−→ S(Ji)

T (2)i−→ Ji

and Ti = T (2)i T (1)

i , where T (1)i is a projection onto the envelope of Ji and T (2)

i is theunique projection from the envelope to the embedded Jordan subalgebra.

• Let Ji be simple with envelope a direct sum of two simple algebras S(Ji) = A1 ⊕A2,then we have

Mdi

T (1)i−→ S(Ji)

T (2),λi−→ Ji

and Ti = T (2),λi T (1)

i , where T (1)i is a projection onto the envelope of Ji and T (2),λ

iis a convex combination (parameter λ ∈ [0, 1]) of projections TA1 : A1 ⊕ 0 → Ji

(TA2 : 0 ⊕A2 → Ji)such that the restriction to the algebraA1 ⊕ 0 (0 ⊕A2) is theunique projection onto the Jordan algebra with simple envelope and TAi is the uniqueextension of this projection.

55


3. Since the enveloping algebras are C*-algebras, projections onto the envelopes are well-known. In particular, if Ji = Mdi ⊗ 1mi , then T : Mdimi → Mdimi a positive, unitalprojection onto J is given by:

T (A ⊗ B) = A ⊗ tr(ρB)1 ∀A ∈ Mdi , B ∈ Mmi (4.1)

Together with the direct sum structure of the map, this completely describes projectionsonto envelopes.

4. If Ji is reversible, then T (2)i = id +α

2 , where α is a positive antiautomorphism such that Ji isthe set of fixed-points of α. In particular, such an α always exists for any simple, reversibleembedding algebra with a simple envelope and T (2)

i is always self-adjoint as a positivemap.

5. If Ji is irreversible, then T (2)i is a projection onto a spin factor.

Let us now outline the strategy of proof: The basic existence and uniqueness results for trace-preserving projections onto Jordan algebras are established in section 4.1 and rely on the Rieszrepresentation theorem. In section 4.2, we cover the case of projections onto the envelope. Here,we first observe that any projection onto a direct sum is a sum of projections of the differentsummands, hence we can restrict to simple factors. Moreover, this restriction is still unique. Inaddition, here we will see that every projection onto a simple Jordan algebra is a concatenation ofa projection onto its envelope and further projections and we observe that the projection onto theenvelope is not necessarily unique, if we have multiplicities.First, we treat the irreversible spin factors (recall that every irreversible representation is also arepresentation of a spin factor) in sections 4.3. Here, we have to distinguish between simple andnonsimple envelopes. The methods for proving uniqueness of the projection onto spin factors withsimple (and non-simple) envelope are very elementary, using only the underlying spin-system toobtain the results. Similarly, we can construct them with elementary tools in section 4.4.In section 4.5, we treat the different reversible summands that arise in our decomposition. We firstconsider the case of simple envelopes, where we prove that there are projections that are a sumof the identity and an antiautomorphism. To this end, we will consider the real algebra R(Jh)generated by the self-adjoint elements of the Jordan algebra and show how the antiautomorphismarises from the decomposition of the complex envelope into R(Jh) + iR(Jh). Then we show thatthe projections are unique by using appendix A.1 to reduce the case to the case of spin factors.Likewise, we reduce the case of Jordan algebras with non-simple envelope to the case of spinfactors (at least partially).Especially the results on projections onto embeddings of spin factorsV2k+1 seem to be unknownin the literature (except for the case V3, which is done in [36] to prove the results stated onreversible algebras). In addition, we give explicit calculations for all projections and also calculatetheir adjoints, which correspond to the fixed point projections of a positive, trace-preserving mapwith a full-rank fixed point. This will later enable us to construct all possible fixed point spaces.

56

Martin Idel Existence and uniqueness of projections onto Jordan algebras

4.1. Existence and uniqueness of projections onto Jordanalgebras

Before delving into this topic, we begin with an abstract existence theorem:

Proposition 4.2. Given a Jordan algebra of self-adjoint elements Jh ⊂ Md(C)h, there exists apositive, linear, unital projection T such that T (Md(C)) = J . Hence T is a positive, linear mapwith T∞ = T.

Proof. We follow the finite dimensional case in [8], where this theorem is stated as lemma 2.3.for spin factors. The idea of the proof is to use the underlying vector space structure togetherwith the Riesz representation theorem to obtain a projection:First note that tr defines the usual inner product on Md via tr : Md × Md → C, (A, B) 7→tr(A†B) =: 〈A, B〉. Since Jh is a real vector space, the restriction of this scalar product to Jh

defines an inner product on Jh. Now for every X ∈ (Md)h, 〈X, ·〉 defines a linear functional onJh, hence by Riesz representation theorem, we obtain:

〈X, A〉 = (T (X), A) ∀A ∈ J

where (·, ·) is the restriction of 〈·, ·〉 to J . This implies that T is well defined on (Md)h and bysesquilinearity of the Riesz-map also well defined onMd. By construction, T ((Md)h) = J (sinceevery finite-dimensional normed space is reflexive) and T 2 = T as well as T is unital, since Jis unital. It remains to be shown that T is a positive map. To this end, assume that T was notpositive. ConsiderMd 3 X ≥ 0 s.th. T (X) 0. Then, since T is Hermitian preserving (as byconstruction, it maps self-adjoint elements to self-adjoint elements), by the spectral theoremT (X) = αP1 − βP2 for some α, β ∈ R+, β , 0 and P1, P2 ∈ J s.th. P1 + P2 = 1 and (P1, P2) = 0.But then:

0 ≤ tr(X†P2) = (T (X), P2) = −β(P2, P2) < 0

which is a contradiction. Hence T must be positive.

We obtain the following important corollary, using the fact that the Riesz representation theoremguarantees uniqueness of the projection constructed in this way:

Corollary 4.3. For every Jordan algebra J ⊂ Md, there exists a unique linear, positive, unitaland trace-preserving projection T onto J .

Proof. Let T be a linear, positive, unital and trace-preserving projection onto J . Given a ∈ Jand x ∈ Md arbitrary, we obtain (since for a, we have equality in the Jordan-Schwarz-inequality):

tr(T (x)†a) = tr(T (x)† ∗ a) = tr(T (x†) ∗ T (a)) = tr(T (x† ∗ a)) = tr(x† ∗ a) = tr(x†a)

where we used the trace-preserving property in the last equality. But then, T must be a projectionof the type constructed in the last proposition. The construction there is unique, since theRiesz-representation theorem states that the element T (x) is unique, hence T is unique.

57

Martin Idel Projections onto direct sums with multiplicity

4.2. Projections onto direct sums with multiplicity

In order to give a characterization of projections onto Jordan algebras, we need to break down theproblem to projections onto the envelopes of simple factors, where the necessary ingredients willbe provided by the next lemmata, which tell us how projections onto envelopes look like:

Lemma 4.4. Given a unital Jordan algebra J :=⊕K

i=1Ji ⊂ Md(C) that is a direct sum ofsimple algebras Ji ⊂ Mdi and given a unital, positive, linear projection T : Md → Md s.th.T (Md) = J and T (J) = J , we can find isometries Vi : Cdi → Cd and unital, positive, linearprojections Ti :Mdi →Mdi s.th.

T (A) =

K∑k=1

VkTk(V†k AVk)V†k (4.2)

Proof. The first two steps essentially follow the proof of the known result for conditionalexpectations in [40] (prop. 1.5). To begin with, consider the enveloping algebras S(Ji) of Ji fori ∈ 1, . . . ,K, i.e. the associative algebras generated by the Ji. These algebras are necessarily*-subalgebras ofMd.We notice that the direct sum

⊕Ki=1Ji ⊂ Md(C) implies in particular

K⊕i=1

Ji ⊂

K⊕i=1

Mdi Ji ⊂ Mdi

for some di ∈ N. Since the generators of Ji lie inMdi , in particular S(Ji) ⊂ Mdi . This meansthat S(J) =

⊕Ki=1 S(Ji).

Now S(Ji) ⊂ Mdi and we have also a decomposition of the Hilbert space Cd =⊕K

i=1Hi (wheredim(Hi) = di), hence we can find isometries Vi : Hi → Cd s.th. V†i Vi = 1di . Furthermore, recallthat 1 ∈ J , hence it is also in S(J), which implies that

∑Ki=1 di = d. With this in mind, define

Pi = ViV†

i , which is a projection ontoHi, hence Pk ∈ S(J) and in particular, it acts as identityonHi and as zero operator onH j for j , i. This implies

∑Ki=1 ViV

†

i =∑K

i=1 Pi = 1

We define the pinching map T0(A) :=∑K

i=1 PiAPi. The basic idea of the proof is now to showthat this map leaves any positive projection onto a Jordan algebra invariant if we apply it beforeapplying T or vice-versa. To this end, consider A ∈ S(J), hence A =

⊕Ki=1 Ai with Ai ∈ Mdi .

Then:

T0(A) = T0(A1 ⊕ A2 ⊕ . . . ⊕ AK) =∑

i

Pi(A1 ⊕ . . . ⊕ AK)Pi

=∑

i

0 ⊕ . . . ⊕ Ai ⊕ . . . ⊕ 0 = A

hence T0 keeps S(J) and therefore J invariant and we have T0T = T .Moreover, consider A± := λPk + λ−1Pl ± (Q + Q†) for λ > 0 and k , l, where ‖Q‖∞ ≤ 1 and

58


PlQPk = Q. Recall that PlMdPk Mdl,dk , i.e. all possible Q have nonzero entries only in thedl × dk block that sends vectors from Hk to Hl (and the transpose of this block), in particular,such Q exist. We want to see that Q = 0 necessarily, which would imply that the non-diagonalblocks always vanish, or equivalently, that TT0 = T .To this end we have A± ≥ 0, by using that(

λ1 QQ† λ−1

1

)≥ 0 ⇔ ‖Q‖∞ ≤ 1

(see [1], theorem 1.3.3 or [40], chapter 5). Hence, T (A±) ≥ 0 by the positivity of T . NowT (A±) = λPk + λ−1Pl ± T (Q + Q∗). Now T (Q + Q∗) = P − N with P,N ≥ 0 the positive andnegative part. Consider first the case A+ then N = 0, since if N , 0, then T (Q + Q∗) possesses anegative eigenvalue −ξ < 0 and as PkNPk + PlNPl = N (here we use the direct sum structure:T (Q + Q†) ∈ J since T is a projection onto J , hence N, P must be invariant under the pinchingmap). Therefore, the corresponding eigenvector ψ ∈ Hk ⊕Hl lies either inHk or inHl. Supposewithout loss of generality that ψ = ψ ⊕ 0. By choosing 0 < λ < ξ/〈ψ|Pk|ψ〉, we can obtain

〈ψ|T (A+)|ψ〉 = λ〈ψ|Pk|ψ〉 + λ−1〈ψ|Pl|ψ〉 − ξ = λ〈ψ|Pk|ψ〉 − ξ < 0

which contradicts the positivity of T . Hence N = 0. Likewise, considering A−, we obtain P = 0,implying T (Q + Q∗) = 0 for all Q s.th. PlQPk = Q (since we can always normalize Q to fulfill‖Q‖∞ ≤ 1). Hence, TT0 = T .Above, we have shown T0TT0 = T , which means we can write T (A) =

∑Kk,l=1 PlT (PkAPk)Pl.

The last step consists in showing that the terms k , l have to vanish. To this end, considerA ∈ Mdk (the k-th direct summand) arbitrary. By abuse of notation, we will also write A forthe respective element inMd. Without loss of generality, by linearity of the map, it suffices toconsider positive elements A (since any element can be decomposed into a sum of four positiveelements). Therefore, we only have to show that PlT (A)Pl = 0 for all l , k. To this end, considerA± = Pk ±A/‖A‖∞ then A± ≥ 0 as above (note that Pk is the identity onMdk ) and T (A±) ≥ 0. Wecan then decompose T (B) into positive and negative part as above and show that since T (Pk) = Pk

as Pk ∈ Jk, the projection to Pl of the positive and negative part of T (B) must vanish.Now define Tk :Mdk →Mdk via Tk(.) := V†k T (Vk.V

†

k )Vk. Then we have the decomposition

T (A) =

K∑k=1

PkT (PkAPk)Pk =

K∑k=1

VkTk(V†k AVk)V†k

Since T is a unital, positive projection, this implies by a straightforward calculation that also theTk must be unital, positive projections:For positivity, note that for all Ak ∈ Mdk with Ak ≥ 0, A = 0 ⊕ . . . ⊕ 0 ⊕ Ak ⊕ 0 ⊕ . . . ⊕ 0 ≥ 0 andwe have

0 ≤ T (A) =

K∑k=1

VkTk(V†k AVk)V†k = VkTk(Ak)V†k

59


hence Tk is positive. Decomposing 1 into the projections Pk shows that Tk must be unital.

Now we wish to treat multiplicities. During the proof, we will need the fact that projections ontoa Jordan algebra with a simple envelope are unique, a result which will only be proven later on inlemma 4.23 for the reversible case and 4.8 for the irreversible case (the proofs there do not needthe following lemma, since the next section is independent of the results of this section).

Lemma 4.5. Let J ⊂ Md be a unital, irreducible representation of a simple Jordan factor.Denote by S(J) its enveloping algebra and assume S(J) is simple. Then S(J) ⊗ N , whereN := S(J)′ the (relative) commutant of S(J). In particular, any projection from Md ontoJ = J ⊗1 is of the form T (A) = T (tr2(A(1⊗ ρ)))⊗1 with ρ ∈ N a state and T a positive, unitalprojection of S(J) to J .

Proof. The proof is clustered from [36]. However, we will need a lot of technical results that wesummarized in A.1 and which are not present in the reference, since they are supposedly commonknowledge in the field.By linearity of T , it suffices to prove T (X ⊗ Y) = tr(Yρ)T (X) ⊗ 1 where X ∈ S(J), Y ∈ NT : S(J)→ J a positive, unital projection and ρ ≥ 0. The proof now proceeds as follows: Weconsider A ⊗ 1 for A ∈ S(J) in step 1 and 1 ⊗ B for B ∈ N in step 2, before we put everythingtogether in step 3.Step 1: First note that T (A ⊗ 1) = A ⊗ 1 for all A ∈ J . Also, by linearity, unitality and positivityof T , we can define a linear, unital and positive map T : S → J such that T (X ⊗ Y) = T (X) ⊗ 1for all X ∈ S. In particular, since T is a projection, T is a projection.Step 2: In this step, we will make use of the concepts introduced in appendix A.1. Now weconsider the case T (1 ⊗ Y) for Y ∈ N . We want to show that T (1 ⊗ Y) = tr(ρY)1 ⊗ 1 for someρ ≥ 0. Consider first Y ≥ 0, then T (1 ⊗ Y) ≥ 0 via positivity. Also,

T (A ⊗ Y) = T ((A ⊗ 1) ∗ (1 ⊗ Y)) = (A ⊗ 1) ∗ T (1 ⊗ Y) ≥ 0

for all 0 ≤ A ∈ J by lemma 2.25. Now we suppose

T (1 ⊗ Y) =

n∑i=1

λiPi

where λi ∈ R+ are the eigenvalues of T (1 ⊗ Y) and Pi their eigenprojections. In particular,∑n

i=1 Pi = 1. If Y , 0, then at least one eigenvalue is nonzero and assuming λ1 , 0 is the biggesteigenvalue, we can normalize the expression to λ1 = 1. We now wish to show that then all theother eigenvalues must necessarily be 1 also, which would imply that T (1 ⊗ Y) ∝ 1 ⊗ 1.Using lemma A.3 that tells us that all spectral projections of an element lie in the Jordan algebra,we can now say that Pi ∈ J , since T (1 ⊗ Y) ∈ J . Also, using section A.1, we can assume thatthe Pi are minimal. Let j = 2, . . . , n. Denote P(1 j) := P1 + P j. Then

(A ⊗ 1) ∗ T (1 ⊗ Y) ≥ 0 ⇒ (P(1 j)(A ⊗ 1)P(1 j)) ∗ (P(1 j)T (1 ⊗ Y)P(1 j)) ≥ 0

60


In particular, using proposition A.9, we can choose A⊗1 to be a minimal projection A⊗1 ≤ P1+P j

different from P1 and P j. Then, using lemma A.7, we know that this projection has the form

A ⊗ 1 = P(1 j)(A ⊗ 1)P(1 j) =

(α1 βBβB† γ1

)with α, γ ≥ 0, α + γ = 1, B†B = BB† = 1 and |β|2 = αγ. But then P(1 j)(A ⊗ 1)P(1 j) = A ⊗ 1 ≥ 0by construction and we have:

P(1 j)(A ⊗ 1)P(1 j) ∗ P(1 j)T (1 ⊗ Y)P(1 j) =

(α1 βBβB† γ1

)∗

(1 00 λ1

)=

α1β(1+λ)

2 Bβ(1+λ)

2 B† γλ1

≥ 0

Using [1], theorem 1.3.3., which states that(A BB∗ C

)≥ 0 ⇔ B∗C−1B ≤ A

hence we obtain:

λαγ ≥

∣∣∣∣∣12β(1 + λ)∣∣∣∣∣2 =

14αγ(1 + λ)2

⇔ λ ≥14

(1 + λ)2

But the last equation is only fulfilled if λ = 1. Since j was arbitrary, every eigenvalue must be1 and hence T (1 ⊗ Y) ∝ 1 ⊗ 1. By linearity and positivity, the proportionality factor must be apositive, linear functional onMd, hence by Riesz representation theorem,

T (1 ⊗ Y) = tr(ρY)1 ⊗ 1

for all positive Y ∈ N. For arbitrary Y , decompose Y into a linear combination of four positiveoperators and use linearity to obtain the same result.Step 3: Note that for A ∈ J , we have T (A ⊗ Y) = T (A ⊗ 1 ∗ 1 ⊗ Y) = T (A ⊗ 1) ∗ T (1 ⊗ Y) =

A ⊗ 1 tr(ρY), using the equality condition in the Jordan Schwarz inequality. This implies thatfor fixed Y , TY : S(J) → J defined via TY (X) ⊗ 1 = T (X ⊗ Y) for all X ∈ S(J) must be apositive projection onto a Jordan algebra with full envelope. Since T (1 ⊗ Y) = tr(ρY)1 ⊗ 1, wehave that TY (1) = tr(ρY)1, hence for Y normalized with respect to tr(ρ.), we have TY is a positiveunital projection. In lemma 4.23 (for the reversible case) and lemma 4.8 (for the irreversiblecase), we will see that such projections are unique, hence TY = T1 = T , or in other words,T (X ⊗ Y) = tr(ρY)T (X) ⊗ 1 for all X ∈ S(J) and for all Y ≥ 0. Using the decomposition ofarbitrary Y ∈ N into four positive operators, the proof is finished.

61

Martin Idel Projections onto spin factors

In passing, we have proven the following

Corollary 4.6 ([36], Lemma 3.1). Let J be a simple Jordan algebra. Let Z ∈ J , Z ≥ 0. If

Z ∗ A ≥ 0 ∀A ∈ J (4.3)

then Z ∝ 1.

4.3. Projections onto spin factors

This section is a partial recollection of the results presented in [10]. The results of this paperapply to the universal embedding of spin factors and contain an error concerning the universalembeddings of even-dimensional spin factors. We will extend the results to cover all embeddings(which means we have to incorporate the embeddings given in proposition 3.8 and fix the error inthe paper). This requires some technicalities.Throughout this section, we always consider the universal embedding of the spin factors fromproposition 3.7 as well as the nonuniversal embeddings in 3.8, which via theorem 3.3 give allequivalence classes of embeddings of spin factors. In particular, given n ∈ N, we will considerthe embedding of a spin factorVk intoM2n for k = 2n− 1, k = 2n and k = 2n + 1, where the firstembedding has an envelope that is a direct sum of two simple factors and the second and thirdembedding have full envelopes. This covers all different cases (up to isomorphy of the envelopes).First define the following elements:

X1 :=(0 11 0

), X2 :=

(0 i−i 0

), X3 :=

(1 00 0

), X4 :=

(0 00 1

),

then the Xi span (M2(C))h as a real vector space, hence they span M2 as a complex vector space.This also implies that

X := Xi1 ⊗ . . . ⊗ Xik i1, . . . ik ∈ 1, . . . , 4 (4.4)

span the spaceM2k , hence we need only consider these elements.Before giving the projections, we will need a preparatory lemma:

Lemma 4.7 ([10]). Let Vk be a spin factor with k = 2n or k = 2n+1. Then for m = 0, 1, . . . , n−1,let X := 1m ⊗ Xim+1 ⊗ . . . ⊗ Xin and T a positive, unital projection ontoVk, we have:

S 2m+2 ∗ (S 2m+2 ∗ P(X)) =12

P(1m+12 ⊗ Xim+2 ⊗ . . . ⊗ Xik ) im+1 = 3, 4 (4.5)

S 2m+1 ∗ P(x) = 0 im+1 = 1, 2 (4.6)

Proof. We follow the proof in [10]. First note that

X3 ∗ σ1 =12

((1 00 0

) (0 11 0

)+

(0 11 0

) (1 00 0

))=

12σ1

62


and likewise for X4. Then, using the Jordan Schwarz equality in lemma 2.25 for fixed pointstwice, we have for Xm+1 = X3 or Xm+1 = X4:

S 2m+2 ∗ (S 2m+2 ∗ T (X)) = S 2m+2 ∗ T (S 2m+2 ∗ X) = T (S 2m+2 ∗ (S 2m+2 ∗ X))

= T (S 2m+2 ∗ (σ(m)2 ⊗ σ1 ∗ 1

m2 ⊗ Xim+1 ⊗ . . . ⊗ Xin))

= T (S 2m+2 ∗ (σ(m)2 ⊗ (σ1 ∗ Xim+1) ⊗ Xim+2 ⊗ . . . ⊗ Xin))

=12

T (S 2m+2 ∗ (σ(m)2 ⊗ σ1 ⊗ Xim+2 ⊗ . . . ⊗ Xin))

=12

T (1m+12 ⊗ Xim+2 ⊗ . . . ⊗ Xin)

which proves equation (4.5). For equation (4.6) first note that X1 ∗ σ3 = X2 ∗ σ3 = 0 sinceX1 = σ1 and X2 = σ2. With this in mind, we have:

S 2m+1 ∗ T (X) = T (S 2m+1 ∗ X)

= T (σ(m)2 ⊗ (σ3 ∗ Xim+1) ⊗ Xim+2 ⊗ . . . ⊗ Xik )

= 0 im+1 = 1, 2

This is equation (4.6). Note that we do not have this equation for S 2k+1, only for all S m with modd and m ≤ 2k − 1. The reason is that if we consider arbitrary embeddings, S 2k+1 has not theform exploited in the lemma, namely S 2k+1 , σ(k)

2 ⊗ σ3 in every embedding. This is differentfrom the proof of the lemma in [10].

With this lemma, we can state the uniqueness result. In contrary to the claims in [10] we only getuniqueness for embeddings with simple envelope, similar to what we will obtain for the reversiblecase:

Lemma 4.8. Let T be a positive, unital projection onto an embedding of the spin factorVk suchthat the envelope of the embedding S(Vk) is simple, then T is unique.

Proof. Following the proof in [10], we will prove that T is trace-preserving, hence T is uniqueby corollary 4.3.Consider an arbitrary positive, unital projection T :M2n →Vk for k = 2n−1, k = 2n or k = 2n+1depending on the embedding ofVk. Consider X of the form above, then X is self-adjoint and wecan write:

T (X) =: a01n2 +

k∑i=1

aiS i (4.7)

with real numbers ai. Since the embeddings of S i are trace-less (and must be traceless in everyembedding, since the trace is unitarily invariant), it suffices to show that a0 = tr(X) with X of theform in equation (4.4) for T to be trace-preserving.

63


To this end, consider first i1 = 1, 2 in X and note that S 1 ∗ T (X) = a0S 1 + a11n2 via the Clifford

relations. But by equation (4.6), S 1 ∗ T (X) = 0 hence a0 = a1 = 0 in this case.Let i3 = 1, 4. Then, using S 1 = σ3 ⊗ 1

(n−1), we have via S 1 ∗ X3 = X3 and S 1 ∗ X4 = −X4:

a0S 1 + a11n2 = S 1 ∗ P(X)

= T (S 1 ∗ X)

=

T (X) if Xi1 = X3

T (−X) if Xi1 = X4

Comparing coefficients with equation (4.7), we obtain:

T (X) =

a01n2 + a0S 1 if X = X3 ⊗ Xi2 ⊗ . . . ⊗ Xin

a01n2 − a0S 1 if X = X4 ⊗ Xi2 ⊗ . . . ⊗ Xin

Using S 2 ∗ S 1 = 0, we calculate:

a01n2 = S 2 ∗ (S 2 ∗ T (X))

= T (S 2 ∗ (S 2 ∗ X))

=12

T (12 ⊗ Xi2 ⊗ . . . ⊗ Xin)

where we used equation (4.5) in the last step. Now we have the same situation as in the preparatorylemma. We now iterate:

• Xi2 = X3 or Xi2 = X4, then

a01n2 =

12

T (12 ⊗ Xi2 ⊗ . . . ⊗ Xin)

= S 4 ∗ (S 4 ∗12

T (12 ⊗ Xi2 ⊗ . . . ⊗ Xin))

=122 T (12

2 ⊗ Xi3 ⊗ . . . ⊗ Xin)

as above using equation (4.5),


a0S 3 = S 3 ∗12

T (12 ⊗ Xi2 ⊗ . . . ⊗ Xin) = 0

using equation (4.6)

Hence we have either a0 = 0 if Xi2 = X1 or Xi2 = X2 or else, we have again a structure of X as inlemma 4.7. Let us write down the final iteration for m = n − 1:

64


• Xin = X3 or Xin = X4, then

a01n2 =

12n−1 T (1n−1

2 ⊗ Xin)

= S 2n ∗ (S 2n ∗12

T (1n−12 ⊗ Xin))

=12n T (1n

2)

as above using equation (4.5),


a0S 2n−1 = S 2n−1 ∗1

2n−1 T (1n−12 ⊗ Xin) = 0

using equation (4.6)

The original paper [10] makes a mistake in this step, claiming that this iteration is true for bothk = 2n and k = 2n − 1, however it does not hold for k = 2n − 1, since S 2n is missing, hence notnecessarily

a01n2 =

12n T (1n

2)

which is needed to have a trace-preserving projection. However, this step is valid for theembedding of the case k = 2n + 1 with simple envelope, since we only need S 2m and S 2m−1 forall m ∈ 1, . . . k. Then we obtain:

a01n2 =

12n T (1n

2) if Xil = X3 or Xil = X4, ∀l ∈ 1, . . . , n0 if Xil = X1 or Xil = X2, for some l ∈ 1, . . . , n

Since T (1n2) = 1

n2 by unitality of T this implies a0 = 0 if Xil = X1 or Xil = X2 for some l and else

a0 = 12n . But then we have:

tr(T (X)) =

1 if Xil = X3 or Xil = X4, ∀l ∈ 1, . . . , n0 if Xil = X1 or Xil = X2, for some il ∈ 1, . . . , n j

On the other hand, we obtain:

tr(X) = tr(Xi1 ⊗ Xi2 ⊗ . . . ⊗ Xin)

= tr(Xi1) tr(Xi2) . . . tr(Xin)

= tr(T (X))

using that X1 and X2 are traceless and X3 and X4 have unit trace. But then, T must be trace-preserving by linearity. Hence corollary 4.3 implies uniqueness of the representation.

65


We already know how the unique projection can be constructed, using Riesz representationtheorem, i.e. we have:

Corollary 4.9. Let T :M2n →Md be a positive projection onto an embedding of the spin factorVk with k = 2n or k = 2n + 1. Denote the spin-system underlying the spin factor by S 1, . . . , S k

then T is given by:

T (A) =tr(A)

2n 1 +

k∑i=1

tr(S †i A)2n S i

Proof. This follows from proposition 4.2 together with corollary 4.3 (the factor 12n is the normal-

ization of the trace).

It would be nice to have a more specific characterization of this positive projection similar tothe reversible case, but this turns out to be rather difficult. However, we will give some othercharacterizations in section 4.4.We have just seen that we have unique projections onto spin factors for simple algebras. We havealso already seen where exactly we have the difference between spin factors with simple envelopeor with envelope that is a direct sum of two envelopes. In this case, there seems to be a choice inT (1n−1 ⊗ X3).The case of representations of spin factors with simple envelopes covers the embedding ofVk

intoM2n for k = 2n and k = 2n + 1, but not the universal embedding for k = 2n − 1. Now let usstudy the projections onto representations of spin factors with an envelope that is a direct sum oftwo simple algebras, i.e. the (universal) representation for k = 2n − 1. This case is similar to thecase of nonsimple envelope for reversible algebras, which we will cover later on. This does notcome as a very big surprise, since the universal embedding of V3 is a case of a reversible algebrawith nonsimple envelope.

Lemma 4.10. Let J be the universal embedding σ0 : V2n+1 →M2n+1 with universal envelopeM2n ⊕M2n . Let T : M2n+1 → M2n+1 be a positive, unital projection onto J . Then there areλ1, λ2 ∈ [0, 1] with λ1 + λ2 = 1 such that for P1 := diag(12n , 0) and P2 := diag(0,12n) we haveT (Pi) = λi1. Then

T (A) = λ1TP1(A) + λ2TP2(A) (4.8)

where TPi(A) =∑2n+1

j=0 tr(PiAPi ∗ PiS jPi)S j/2n+1.In other words, the restriction of the embedding of the spin-system S j

2n+1j=0 to PiM2n+1 Pi is an

embedding ofV2n+1 intoM2n ⊕ 0 (or 0 ⊕M2n). Then TPi |Pi is the unique projection onto thisembedding and TPi is the natural extension to the embedding intoM2n+1 . Then T is a convexcombinations of the unique projections corresponding to the two possible restrictions P1 and P2.

Proof. Let us first outline the strategy of proof. In the first step, we will see that the off-diagonalblocks P1MdP2 + P2MdP1 vanish. Then we prove the claim that T (Pi) ∝ 1 in step 2, before

66


we can finally construct TPi in the third step, thereby reducing the problem to the problem ofprojections onto spin factors with simple envelope that we have already covered.Step 1: Claim: T (P1XP2 + P2XP1) = 0 for all X ∈ M2n+1 .We use a different terminology of the proof of lemma 4.8, albeit the ideas are similar. Recall that1 =: σ0, σ1, σ2, σ3 span (M2)h. Then the space P1XP2 + P2XP1 | X ∈ M2n+1 is spanned by

X := σi1 ⊗ . . . ⊗ σin ⊗ σ1/2

hence we need to see that T (X) = 0 for all X of this type.Let T (X) = a01 +

∑2n+1i=1 aiS i, since T (X) ∈ J . Then, since S 2n+1 = σn

2 ⊗ σ3 and σ3 ∗ σ1/2 = 0,we observe:

S 2n+1 ∗ T (X) = T (S 2n+1 ∗ X) = T ((σ2 ∗ σi1) ⊗ . . . ⊗ (σ3 ∗ σ1/2)) = 0

using the equality in the Jordan-Schwarz inequality for TS n+1 = S n+1, lemma 2.25. On the otherhand, S 2n+1 ∗ (a01 +

∑2n+1i=1 aiS i) = a0S 2n+1 + a2n+11, hence a0 = a2n+1 = 0.

Now consider first σi1 = σ1. Then

S 2 ∗ (S 2 ∗ T (X)) = T (S 2 ∗ (S 2 ∗ X))

= T (S 2 ∗ (σ1 ∗ σ1 ⊗ σi2 ⊗ . . . ⊗ σin ⊗ σ1/2))

= T (σ1 ⊗ σi2 ⊗ . . . ⊗ σ1/2)

= T (X)

But also S 2 ∗ (S 2 ∗ (a01 +∑2n+1

i=1 S i)) = a01 + a2S 2. Since we already knew that a0 = 0, we haveT (X) = a2S 2.Now a21 = S 2 ∗ T (X) = T (S 2 ∗ X) = T (1 ⊗ σi2 ⊗ . . . ⊗ σ1/2). But then

a2S 2n+1 = S 2n+1 ∗ T (1 ⊗ σi2 ⊗ . . . ⊗ σ1/2) = 0

as σ3 ∗ σ1/2 = 0, which implies a2 = 0, hence T (X) = 0.Next consider σi1 = σ3, then, using S 1 instead of S 2 we obtain T (X) = a01+ a1S 1. Furthermore,a0 = 0 is known and calculating 0 = S 2n+1 ∗ (S 1 ∗ T (X)) = a1S 2n+1 we obtain a1 = 0, henceT (X) = 0.For σi1 = σ0, we have already seen that T (X) = 0, since a01 + a2S 2 = S 2 ∗ (S 2 ∗ T (X)) = T (X)and a2S 2n+1 = S 2n+1 ∗ T (X) = 0, hence T (X) = 0.The last case to consider is σi1 = σ2. In this case, we can iterate. Let σi2 = σ1, then, usingS 4 = σ2 ⊗ σ1 ⊗ 1

n−1, we have

a01 + a4S 4 = S 4 ∗ (S 4 ∗ T (X))

= T (S 4 ∗ ((σ2 ∗ σ2 ⊗ σ1 ∗ σ1 ⊗ σi3 ⊗ . . . ⊗ σin ⊗ σ1/2)))

= T (S 4 ∗ (1 ⊗ 1 ⊗ σi3 ⊗ . . . ⊗ σin ⊗ σ1/2))

= T (σ2 ⊗ σ1 ⊗ σi3 ⊗ . . . ⊗ σin ⊗ σ1/2)

67


= T (X)

Then, as above we know that a0 = 0 and a4 = 0 via a4S 2n+1 = S 2n+1 ∗ T (X) = 0. Similarly, thecases σi2 = σ0/3 are handled and for σi2 = σ2, we can iterate.The last case, which is not covered by the iteration, is

X = σn2 ⊗ σ1/2

Clearly, S 2n+1 ∗ T (X) = 0 as above, but in addition, as σ2 ∗ σ1/3 = 0, we also have S i ∗ T (X) = 0for all i = 1, . . . , 2n. Hence T (X) = 0.In summary, T (X) = 0 for all X = σi1 ⊗ . . . σin ⊗ σ1/2, i.e. T (P1XP2 + P2XP1) = 0.Step 2: T (Pi) = λi1 for some λi ∈ [0, 1] such that λ1 + λ2 = 1.By definition, P1 = σn

0 ⊗ ε11 and P2 = σn0 ⊗ ε22, where we define

ε11 =

(1 00 0

), ε22 =

(0 00 1

)Then T (Pi) = a01 +

∑2n+1i=1 aiS i with real coefficients ai as above. For j = 1, . . . , 2n we obtain:

a01 + a jS j = S j ∗ (S j ∗ T (X)) = T (S j ∗ (S j ∗ X)) = T (X)

since S j = σ j1 ⊗ . . . ⊗ σ jn ⊗ 1 for some j1, . . . , jn. Comparing coefficients then shows thatT (Pi) = a(i)

0 1 for i = 1, 2 and some a(i)0 ∈ R. Furthermore, since T is positive, a0 ≥ 0 and since T

is unital, 1 = T (1) = T (P1) + T (P2) = (a(1)0 + a(2)

0 )1, which proves the claim.Step 3: Decomposition and reduction to the case of a simple envelope:Using Step 1, we can see that for any X ∈ M2n+1 and using T (X)Pi = T (X) ∗ Pi, since Pi is in thecentre of the envelope of the image of T :

T (X) = T ((P1 + P2)X(P1 + P2)) = T (P1XP1) + T (P2XP2)

= T (P1XP1)P1 + T (P1XP1)P2 + T (P2XP2)P1 + T (P2XP2)P2

Now we want to have a look at T (PiXPi)Pi.Using S jPi = PiS j for all j = 1, . . . , 2n + 1, since Pi ∈ C(S(J)), T (S i) = S i and the lemmaabout the conditions for equality in the Jordan-Schwarz-inequality (lemma 2.25), we obtain

T (PiS jPi)Pi = T (Pi ∗ S j)Pi = (T (Pi) ∗ S j)Pi = λiS j ∗ Pi

= λiPiS jPi(4.9)

Now define Ti : M2n → M2n via T (X) = λ−1i T (X ⊕ 0)Pi|M2n (where we set λ−1

i = 0 if λi = 0),where X ⊕ 0 is just the decomposition according to the embedding ofM2n → PiM2n+1 Pi withX 7→ X ⊕ 0. Then we have Ti(1) = λ−1

i T (Pi)Pi|M2n = 1, hence T is unital. Furthermore, since

68


X 7→ PiXPi is a positive map, restrictions are positive and T is positive, T is also positive.Finally, Ti(Ti(X)) = Ti(X) for all X ∈ M2n since by definition,

T (X) = λ−1i T (X ⊕ 0)Pi|M2n = λ−1

i

2n+1∑j=0

a jS jPi|M2n

= λ−1i

2n+1∑j=0

a jPiS jPi|M2n

Via equation (4.9), if we define S j = PiS iPi|M2n , we have in particular

T (S j) = S j

hence T is a projection onto the algebra spanned by S j2n+1j=0 . Since by definition S j = S j ⊗ 1

for j = 1, . . . , 2n and S 2n+1 = S 2n+1 ⊗ σ3, the algebra spanned by the S j is the embedding of thespin factorV2n+1 intoM2n from proposition 3.8. But then, T is a positive, unital projection ontoa spin factor with simple envelope, hence it is unique by lemma 4.8. Consequently

T (X) =

2n+1∑j=0

tr(X ∗ S j)S j

This basically finishes the proof, if we observe with the above calculations that this implies

λ−1i T (PiXPi)Pi =

2n+1∑j=0

tr(PiXPi ∗ PiS jPi)PiS iPi

⇒ λ−1i T (PiXPi) =

2n+1∑j=0

tr(PiXPi ∗ PiS jPi)S i

Hence T (X) =∑2

i=1 λi(λ−1i T (PiXPi)) = λ1TP1(X) + λ2TP2(X) defining TPi as in the proposition.

Corollary 4.11. Consider the universal embedding σ0 of the spin factorV2n+1 intoM2n+1 as in

proposition 3.7. Let T ∗ be a positive unital projection onto Vσ02n+1 such that T has a full-rank

fixed point. Then, given the notation of lemma 4.10

T∞(A) =

2∑o,p=1

2n+1∑j=0

λoPoS jPo tr((PpAPp) ∗ (PpS jPp))/2n+1

where λ1 + λ2 = 1 and λ1, λ2 ∈ (0, 1).

69

Martin Idel Construction of positive projections onto spin factors

Proof. We have to compute T from T ∗ given in lemma 4.10. Using cyclicity of the trace andP2

o = Po we obtain (for A, B ∈ M2n+1 arbitrary):

tr

A†2∑

o,p=1

2n+1∑j=0

λpPoS jPo tr(PpBPp ∗ PpS jPp)/2n+1

=

2∑o,p=1

2n+1∑j=0

λp trPoA†PoPoS jPo tr(PpBPpS j)/2n+1

=

2∑o,p=1

2n+1∑j=0

λp trtr(PoA†PoPoS jPo)PpBPpS j/2n+1

=

2∑o,p=1

2n+1∑j=0

λp trPpS jPp/2n+1 tr(PoA†PoPoS jPo)B

= tr

2∑

o,p=1

2n+1∑j=0

λpPpS jPp/2n+1 tr(PoAPoPoS jPo)

†

B

using that the Pi and S j are self-adjoint. But then we have the projection as claimed. Now ifλ1 = 0, then P2T∞(A)P2 = 0 for all A, hence T∞ cannot have a full-rank fixed point. Therefore,λ1 , 0 and likewise, λ2 , 0.

This finishes the preliminary considerations about projections onto irreversible subalgebras. Inprinciple, we have also seen that a projection onto a subalgebra with envelope that is isomorphic toa sum of two simple algebras is given by a concatenation of (a) a projection onto the envelope and(b) a convex combination of the obvious extensions of the projection from one of the summandsto the Jordan algebra restricted to that summand. Note however, that we have not fully proven thisfor all possible embeddings of the envelopes, yet; the last step will be done in the next section.

4.4. Construction of positive projections onto spin factors

We can now collect our findings from the previous section and the knowledge of representationtheory to give a complete description of projections onto irreversible Jordan algebras. We needthese to construct and characterize the fixed point spaces of trace-preserving maps. Usingtheorem 3.3 and proposition 3.4, we know we only have to cover spin factors with either simple ornonsimple envelope, which gives two cases. In addition, we will give different characterizationsof the (unique) projection onto a spin factor with simple envelope.

70


Projections onto representations of spin factors with simple envelope

For K ∈ N consider the Jordan algebra

J := U

K⊕i=1

Jspinki⊗ 1mi

U† ⊂ Md (4.10)

where U ∈ Md a unitary and Jspinki⊂ M2ni with ki = 2ni or ki = 2ni + 1 and Jspinki

=

S (Vσ0ki

)S ∗ with the universal embedding σ0 of the spin factor Vki as in proposition 3.7 forki = 2ni and in proposition 3.8 for ki = 2ni + 1, and unitary S ∈ M2ni . We can push the unital Sinto U to obtain U:

U

K⊕i=1

S i(Mdi(R)σ0 + iMdi(R)σ0)S †i ⊗ 1mi

U†

= U

K⊕i=1

(S i ⊗ 1mi)[(Mdi(R)σ0 + iMdi(R)σ0) ⊗ 1mi

](S †i ⊗ 1mi)

U†

= U((S 1 ⊗ 1m1) ⊕ . . . ⊕ (S K ⊗ 1mK )) K⊕i=1

(Mdi(R)σ0 + iMdi(R)σ0) ⊗ 1mi

((S †1 ⊗ 1m1) ⊕ . . . ⊕ (S †K ⊗ 1mK ))U†

= U

K⊕i=1

(Mdi(R)σ0 + iMdi(R)σ0) ⊗ 1mi

U†

with a different unitary U.In addition, the form of J leads to a decomposition of the underlying Hilbert space

Cd =

K⊕k=1

Hk =

K⊕k=1

(Hk,1 ⊗Hk,2) K⊕

k=1

(Cdk ⊗ Cmk ) (4.11)

Denote by Vk : Hk → Cd the corresponding isometries and by trk,2 the partial trace over theHilbert space with label k, 2 and by trk the partial trace over the Hilbert space with label k as inthe decomposition of equation (4.11). Then define:

S ki,σ0j := S σ0

j ⊗ 1mi ⊂ Jspinki ⊗ 1mi , j = 0, . . . , ki

the embedding of the spin-system into J resulting from the universal embedding in propositions3.7 and 3.8, where we again let S σ0

0 denote the identity.

71


Proposition 4.12. Let J be a Jordan algebra as in equation (4.10) and denote by Vk theisometries and by tri, tri,k the partial traces obtained from equation (4.11). Any unital, positive,linear projection T ∗ ofMd onto J has the form:

T ∗(A) =

K∑i=1

ki∑j=0

Vi

tri[(

tri,2(V†i AVi(12ni ⊗ ρi)) ⊗ 1mi

)S ki,σ0

j

]2nimi

S ki,σ0j

V†i (4.12)

where ρi ∈ Mmi are states.

Proof. First note that the enveloping algebra of any summand Jspinkiis isomorphic to a simple

matrix algebra. Using section 4.2, we can see that T is a concatenation of projections Tk :S(Jspinki

) → Jspinkiand projections onto the envelope. The statement is the concatenation

written out and simplified.

Before we calculate the adjoint map, we will prove a well-known technical lemma, which tells ushow to compute the adjoint of partial traces:

Lemma 4.13. Let A, B ∈ Md ⊗Md′ , C ∈ Md and D, E ∈ Md′ , then

tr(A[tr2(B(C ⊗ D)) ⊗ E]) = tr([tr2((C ⊗ E)A) ⊗ D]B) (4.13)

Likewise, for A, B,C,D as above and E ∈ Md, we have

tr(A[E ⊗ tr1(B(C ⊗ D))]) = tr([C ⊗ tr1((E ⊗ D)A)]B) (4.14)

Proof. We write the calculation out, starting with the first equation. To this end, let |i1〉 be a basisofMd and |i2〉 a basis ofMd′ . The trick is to view the traced out system as if we had a biggersystemMd ⊗ Md′ ⊗ Md′ , where the third system is traced out and has the same basis as thesecond. We denote it by |i3〉. Furthermore, if B =

∑i jkl bi jklEi j ⊗ E′kl in the chosen basis with

bi jkl ∈ C and matrix units Ei j, E′kl. Then define:

B =∑i jkl

bi jklEi j ⊗ 1d′ ⊗ E′kl ∈ Mdd′dprime

Then we have:

tr(A[tr2(B(C ⊗ D)) ⊗ E]) = 〈i1|〈i2|A〈i3|B(C ⊗ D)|i3〉 ⊗ E|i1〉|i2〉

= 〈i1|〈i2|〈i3|(A ⊗ 1)B(C ⊗ E ⊗ D)|i1〉|i2〉|i3〉

= 〈i1|〈i2|〈i3|[(C ⊗ E)A ⊗ D]B|i1〉|i2〉|i3〉

= 〈i1|〈i3|[〈i2|(C ⊗ E)A|i3〉 ⊗ D]B|i1〉|i3〉

= tr([tr2((C ⊗ E)A) ⊗ D]B)

using the cyclicity of the trace.For the second equation, we do a similar computation. To this end we consider the system as

72


above with a basis |i2〉 ofMd and a basis |i3〉 ofMd′ and consider the bigger system, where thefirst system is traced out and has the same basis as the second. We denote it |i1〉. Furthermore,define B as before (with 1d inserted instead of 1d′) and we have:

tr(A[E ⊗ tr1(B(C ⊗ D))]) = 〈i2|〈i3|A[E ⊗ 〈i1|B(C ⊗ D)|i1〉]|i2〉|i3〉

= 〈i1|〈i2|〈i3|(1 ⊗ A)B(C ⊗ E ⊗ D)|i1〉|i2〉|i3〉

= 〈i1|〈i2|〈i3|C ⊗ [(E ⊗ D)A]B|i1〉|i2〉|i3〉

= 〈i1|〈i3|[C ⊗ 〈i2|(E ⊗ D)A|i2〉]B|i1〉|i3〉

= tr([C ⊗ tr1((E ⊗ D)A)]B)

This will help us in commuting the adjoint map:

Corollary 4.14. Let T be a positive, trace-preserving map with a full-rank fixed point such thatthe fixed point algebra of T ∗ is given by J in equation (4.10). Denote by Vk the isometries andby tri, tri,k the partial traces as obtained from equation (4.11). then its fixed point projection T∞has the form:

T∞(A) =

K∑i=1

ki∑j=1

12ni

Vi

(S σ0

j ⊗ ρi tri(Sk j,σ0j V†i AVi)

)V†i

where ρi ∈ Mmi > 0 is a state.

Proof. Given the expression for T∞, the ρi must necessarily be full-rank, since otherwise thefixed points are not full-rank.We have to find the adjoint to T ∗ given in the proposition. First rewrite: In particular, to givedifferent characterizations of what happens:

T ∗(A) =

K∑i=1

ki∑j=0

Vi

tri[(


)S ki,σ0

j

]2nimi

S ki,σ0j

V†i

=

K∑i=1

ki∑j=0

Vi

tri[tri,2(V†i AVi(12ni ⊗ ρi))S

σ0j

]2ni

S ki,σ0j

V†i

=

K∑i=1

ki∑j=0

Vi

S σ0j ⊗

tri,1[(

tri,2(V†i AVi(12ni ⊗ ρi))Sσ0j

)⊗ 1mi

]2ni

V†i

Now calculate the adjoint:

tr

B†K∑

i=1

ki∑j=0

Vi

S σ0j ⊗

tri,1[(


)⊗ 1mi

]2ni

V†i

73


=

K∑i=1

ki∑j=0

12ni

trV†i B†Vi

(S σ0

j ⊗ tri,1[(


)⊗ 1mi

])(∗)=

K∑i=1

ki∑j=0

12ni

tr1 ⊗ tri,1(S ki,σ0

j V†i B†Vi)((


)⊗ 1mi

)=

K∑i=1

ki∑j=0

12ni

tr(

S σ0j ⊗ tri,1(S ki,σ0

j V†i B†Vi)) (


)(+)=

K∑i=1

ki∑j=0

12ni

tr(

tri2

[S σ0

j ⊗ tri,1(S ki,σ0j V†i B†Vi)

]⊗ ρi

)V†i AVi

= tr

K∑

i=1

ki∑j=1

12ni

Vi

(S σ0

j ⊗ ρi tr(S k j,σ0j V†i BVi)

)†V†i A

where in (∗) we used equation (4.14) with C = D = 1, E = S σ0

j and in (+) we used equation(4.13) with C = 1, D = ρi, E = 1.

Projections onto representations with nonsimple envelopeFor K ∈ N consider the Jordan algebra

J := U

K⊕i=1

Jspinki

U† ⊂ Md

where U ∈ Md is a unitary and Jspinki⊂ M2ni m(1)

i +2ni m(2)i

with ki = 2ni + 1 and S(Jspinki, σ0) =

M2ni ⊗1m(1)i⊕M2ni ⊗1m(2)

iwith the universal embedding σ0 of the spin factorVki as in proposition

3.7. Again, all unitaries are gathered in the unitary U, such that this expression contains allpossible embeddings of spin factors.Recall the embedding of the spin-system intoM2ni m(1)

i +2ni m(2)i

that led to the universal embedding:

S ki,σ01 := σ3 ⊗ 1

(ni−1) ⊗ diag(1m(1)i,1m(2)

i)

S ki,σ02 := σ1 ⊗ 1

(ni−1) ⊗ diag(1m(1)i,1m(2)

i)

...

S ki,σ02ni

:= σ(ni−1)2 ⊗ σ1 ⊗ diag(1m(1)

i,1m(2)

i)

S ki,σ02ni+1 := σ(ni)

2 ⊗ diag(1m(1)i,−1m(2)

i)

Furthermore, define S ki,σ0,( j)i to be the restriction toM2ni m(1)

ifor j = 1 and toM2ni m(2)

ifor j = 2.

Then S ki,σ0i = S ki,σ0,(1)

i ⊕ 0 + 0 ⊕ S ki,σ0,(2)i .

74


Define

P(1)i := diag(12ni m(1)

i, 0)

P(2)i := diag(0,12ni m(2)

i)

⇒ P(1)i + P(2)

i = 1M2ni m(1)

i +2ni m(2)i

=: Pi

(abuse of notation: We restrict to the i-th summand in the direct sum in J , in principle, we wouldhave Pi = diag(0, . . . , 0,1M

2ni m(1)i +2ni m(2)

i

, 0, . . . , 0)).

The decomposition of J leads to a decomposition of the underlying Hilbert space similar toequation (4.11).

Cd =

K⊕i=1

Hi =

K⊕i=1

(H

(1)i,1 ⊗H

(1)i,2 ⊕H

(2)i,1 ⊗H

(2)i,2

)

K⊕i=1

(Cni ⊗ Cm(1)i ⊕ Cni ⊗ Cm(2)

i )

(4.15)

Denote by Vi : Hi → Cd the corresponding isometries. In particular, Vi(Vi)† = 1M2ni m(1)

i +2ni m(2)i

=

Pi. Then we have isometries V (1)i and V (2)

i for all i ∈ 1, . . . ,K such that (V ( j)i )†V ( j)

i = P( j)i and

V (1)i (V (1)

i,1 )† = 1M2ni⊗Mm(1)i

, V (2)i (V (2)

i )† = 1M2ni⊗Mm(2)i

. Finally, denote by tr( j)i,k the partial trace over

the Hilbert spaceH ( j)i,k and by tr(i)

i the trace overH ( j)i,1 ⊗H

( j)i,2 as in the decomposition of equation

(4.15).

Proposition 4.15. Given the setting described in this paragraph, for every i = 1, . . . ,K, thereare λ(1)

i , λ(2)i ∈ [0, 1] such that for every positive, unital projection T : Md → J we have

T (P(1)i ) = λ(1)

i Pi and T (P(2)i ) = λ(2)

i Pi and λ(1)i + λ(2)

i = 1. Then any positive, unital projection T ∗

onto J is given by:

T (A) =

K∑i=1

ki∑

j=0

λ(1)i Vi

tr(1)

i

[(tr(1)

i,2

[V (1)∗

i AV (1)i

(12ni ⊗ ρ

(1)i

)]⊗ 1m(1)

i

)S ki,σ0,(1)

j

]2nim(1)

i

S ki,σ0j

V†i

+

ki∑j=0

λ(2)i Vi

tr(2)

i

[(tr(2)

i,2

[V (2)∗

i AV (2)i

(12ni ⊗ ρ

(2)i

)]⊗ 1m(2)

i

)S ki,σ0,(2)

j

]2nim(2)

i

S ki,σ0j

V†i

(4.16)

where the ρ(1)i , ρ(2)

i are states.

Proof. The proof is basically identical to the proof for the simple case lemma 4.10, where we useproposition 4.12 to write out the projection onto the simple factors. Only the first step, where we

75


need to see that the off-diagonal elements vanish, might be different, since now we have to have alook at

X = σi1 ⊗ . . . ⊗ σin ⊗

02ni m(1)i

A

A† 02ni m(2)i

with A arbitrary. Furthermore, we know

S ki,σ02ni+1 = σn

2 ⊗ diag(12ni m(1)i,−12ni m(2)

i)

hence one can easily see X ∗ S 2ni+1 = 0 for all X of this form. This is the only difference in theproof.

Corollary 4.16. Consider the setting described in this paragraph as for proposition 4.15. Givena positive, trace-preserving map T with a full-rank fixed point such that the fixed point algebra ofT ∗ is given by J , then its fixed point projection T∞ has the form:

T∞(A) =

K∑i=1

ki∑j=0

2∑o,p=1

λ(o)i

2niV (o)

i

(S ki,σ0,(o)

j

(12ni ⊗ ρ

(o)i

))V (o)∗

i tr(p)i

[V (p)∗

i AV (p)i S ki,σ0,(p)

j

]where λ(1)

i , λ(2)i ∈ (0, 1), λ(1)

i + λ(2)i = 1 and ρ(1)

i > 0, ρ(2)i > 0 are states.

Proof. Decomposing ViSki,σ02ni+1V∗i =

∑2o=1 V (o)

i S ki,σ0,(o)2ni+1 V (o)∗

i , the corollary follows from calculat-ing T from T ∗ in equation (4.16). Comparison with corollary 4.11 gives the result. The fact thatρ(o)

i are positive definite is necessary for T∞ to have a full-rank fixed point, as is the conditionλ(1)

i ∈ (0, 1) (see cor. 4.11).

Specific construction of the projection

Let us specify this construction first for the universal embeddings. Unfortunately, an explicitconstruction does not give results as nice as for the reversible embeddings. In fact, closedexpressions are very lengthy. We will give two different descriptions of the projection, but weneed a preparatory lemma first.For computational convenience, we will now start counting the matrix entries from 0 instead of 1until the end of this section.

Lemma 4.17. Let Ei j ∈ M2k be a matrix unit, i.e. the (i, j)-th entry is 1 and the all other entriesare zero. Denote the matrix units ofM2 by εi j. Then we have:

Ei j = εi mod 2 j mod 2 ⊗ εb i2c mod 2

⌊ j2

⌋mod 2 ⊗ . . . ⊗ ε

⌊i

2k

⌋mod 2

⌊ j2k

⌋mod 2 (4.17)

where b.c means that we round to the lower number.

76


Proof. This follows from the embedding of the matrix algebras into each other as we fixed it inthe definition of proposition 3.7.

Note that

ε00 =12 + σ3

2, ε11 =

12 − σ3

2

ε01 =σ1 − iσ2

2, ε10 =

σ1 + iσ2

2

Now we define ⊕ to mean bitwise addition, i.e. for i, j ∈ N0 with i =∑∞

n=0 in2n and j =∑∞

n=0 jn2n

we have i⊕ j =∑∞

n=0((an + bn) mod 2)2n.In addition, consider the alphabet Σ = 0, 1 and words over Σ are just strings of letters, then j isa word over Σ in its binary representation, i.e. jn−1 . . . j0 with ji ∈ Σ for all i. Given two wordsv,w with letters in Σ, then let |v| denote the length of the word, i.e. the number of letters. Then vwis another word with length |vw| = |v| · |w|.

Proposition 4.18. Consider the setting and definitions described in this paragraph. LetVk bethe universal embedding of the spin factor with k = 2n, k = 2n− 1 or the embedding of k = 2n + 1into the envelope of k = 2n. For every positive integer i, we consider its binary representation asa word over Σ with length n and letters i j, j = 0, . . . , n − 1.Then T :M2n →Vk, the unique positive, unital projection onto the spin factor, is given by

T (A) =

n−1∑j=0, k=0

j⊕k=w0w1, |w0w1 |=nw0=0...0, w1=1...1

[(i|w1 |−1(−1)

∑|w1 |−2l=0 kl

)S 2|w1 |/2

n

+

((−1) j|w1 | i|w1 |(−1)

∑|w1 |−1l=0 kl

)S 2|w1 |+1/2n + δ|w1 |0(1 + i)S 0/2n

]A jk

(4.18)

where we set jn = 0 to have the right prefactor of S 2n+1, we set S 2n+1 := 0 for k = 2n and likewiseS 2n := 0 and S 2n+1 := 0 for k = 2n − 1 such that the above definition is well-defined for all threecases.

The last term in the sum for S 0 stems from the fact that we have the wrong prefactor of 1otherwise (i instead of 0).

Proof. For the cases k = 2n and k = 2n − 1, for every l = 1, . . . , 2(n − 1) + 2, we need to evaluatetr(S iE jk). To this end recall

S 2m+2 = σ(m)2 ⊗ σ1 ⊗ 1

(n−1−m)

S 2m+1 = σ(m)2 ⊗ σ3 ⊗ 1

(n−1−m)

77


for all m = 0, . . . , n − 1.We have the binary representations j = jn−1 . . . j0 and k = kn−1 . . . k0. Then observe in particular:

E jk = ε j0k0 ⊗ . . . ⊗ ε jn−1kn−1

and thus:

tr(E jkS 2m+2/1) = tr(ε j0k0σ2) . . . tr(ε jm−1km−1σ2)

tr(ε jmkmσ1/3) tr(ε jm+1km+1σ0) . . . tr(ε jn−1kn−1σ0)(4.19)

Now we have to consider the two cases 2m + 2 and 2m + 1 separately.Case 1: Consider 2m + 2:We can see that the expression in equation (4.19) does not vanish if and only if jl , kl for l ≤ mand jl = kl for l > m, since tr(σ0) = 2 and tr(σi) = 0 for i = 1, 2, 3. Recall ε01 = 1

2 (σ1 − iσ2) andε10 = 1

2 (σ1 + iσ2). This implies that for the first m − 1 factors in equation (4.19), we get a factorof i, i.e. im in total, and for every factor with ε01 we obtain a factor of (−1) (for the other termswe do not get any prefactor), hence we obtain:

(4.19) = im(−1)#ε01

if it is not zero. Now #ε01 =∑m−1

l=0 kl, which gives the sum in the exponent in the expression ofthe projection.Case 2: Consider 2m + 1:Similarly, we have that (4.19) does not vanish iff jl , kl for l ≤ m − 1 and jl = kl for l > m − 1.Likewise, we see that the prefactors from the first m − 1 terms in the expression (4.19) is given by

(4.19) = im(−1)#ε01

while the prefactor for the last n − 1 − m terms is 1 each and

tr(ε jmkmσ3) =

1 if jm = km = 0−1 if jm = km = 1

Choice of m:The prefactors just calculated look similar to the ones in the proposition, however, we still have tosee which m do not vanish. To this end, observe that above we always had the conditions jl = kl

for l ≥ p and jl , kl for l < p for some integer p, otherwise the expression (4.19) must vanish.

This expression translates to j⊕k =

l≥p︷︸︸︷0 . . . 0

l<p︷︸︸︷1 . . . 1. In other words, j⊕k does not contain the string

10.Therefore, only terms

j⊕k = w0w1 with w0 = 0 . . . 0 w1 = 1 . . . 1

78


give nonzero terms for T (E jk). Also, m = |w1|, hence the prefactors as given in the proposition.Since |w1| ∈ 0, . . . n by definition, in order to have a well-defined expression, we need to setkn to have a well-defined expression. Also, if S 2n or S 2n+1 are not defined, because we lookat expressions for k = 2n or k = 2n + 1, we can set them = 0 to still have our well-definedexpression.Now in order to obtain the prefactor of S 2n+1 for the nonuniversal embedding, recall that

S 2n+1 = εS 1 . . . S 2n

for ε ∈ ±1,±i. A quick calculation shows that S 2n+1 = σ(n)2 . Thus we have the same situation

as in equation (4.19). The prefactor must be as given in the case 2m + 1 for the case ε00. Hencesetting kn = 0 gives us the correct prefactor.Finally, we need the prefactor of S 0 = 1 which is just tr(Ei j). Now evaluating the expression inequation (4.18), we would obtain tr(Ei j) = i if i = j and 0 else. Obviously, we want tr(Ei j) = 1 ifi = j and 0 else. Since i⊕ j = 0 iff i = j, in this case w1 is the empty word and δ|w1 |0 produces thegiven result. Hence the projection must be as claimed.

The above characterization is explicit and easy to compute in principle. It also shows us severalfeatures of the projection as such, namely that it sends matrix units to a sum of two spinsymmetries. However, it is not very elegant and its use of bitwise addition might be unsuitablefor further study. There is another characterization of the projections onto universal embeddings,which is given in [10]. We cite it here for the reader’s convenience:

Proposition 4.19 ([10], Theorem 2.4 and onwards). For a positive integer k, given the universalembedding ofVn intoM2k(C), we can define the unique positive projection T ontoVn recursively:Let εi j be the matrix units inM2 as above and define

y := εi1 j1 ⊗ . . . ⊗ εik jk ∈ M2k

Define the nonnegative integer m = m(x) as the cardinal number of the set (il, jl)|(il, jl) = (0, 1).Then for k = 1, 2, . . . we obtain the recursive definition:

(i)

T2k+2(y ⊗ ε00) =

T2k(y) ⊗ 1

2 if some il, jl , 0, 1

T2k(y) ⊗ 1

2 +ik(−1)m

2k+1 S 2k+1 if all il, jl , 0, 1

(ii)

T2k+2(y ⊗ ε11) =

T2k(y) ⊗ 1

2 if some il, jl , 0, 1

T2k(y) ⊗ 1

2 +ik(−1)m+1

2k+1 S 2k+1 if all il, jl , 0, 1

79

Martin Idel Projections onto reversible Jordan algebras

(iii)

T2k+2(y ⊗ ε01) = T2k+2(y ⊗ ε10) =

0 if some il, jl , 0, 1ik(−1)m

2k+1 S 2k+2 if all il, jl = 0, 1

(iv)

T2k+1(y ⊗ εi j) =

T2k+2(y ⊗ εi j) if i = j0 if i , j

(v)

T2(ε00) = ε00 T2(ε01) = T2(ε10) =12σ1 T2(ε11) = ε11

Proof. We need to see that this description is equivalent to the other description, which can beseen by inspection. We will give a sketch of the crucial elements:First notice that the factors of 1

2k add up to give 12n if every summand and amount to the normal-

ization of S i in equation (4.18). Item (iii) tells us (taken together with the first case of (i) and(ii)) that y must consist only of ε01 or ε10, hence, looking at the binary representation of i and jsuch that Ei j = εi0 j0 ⊗ . . . ⊗ εin−1 jn−1 , all il must first be different from jl and then they must be thesame, i.e. i⊕ j + 1 = w0w1 with w0 = 0 . . . 0 and w1 = 1 . . . 1 as in equation (4.18).The second cases of (i) and (ii) correspond exactly the case 2m + 1, where we obtained anadditional sign for ε11 when the trace did not vanish. Furthermore, m =

∑kl=0 jl by definition of m,

hence the prefactors are the same. The first case of (i), (ii), (iii) and (iv) together also show that aslong as the last elements in the tensor product y ⊗ εi j are ε11 or ε00, the result of T2k+2(y ⊗ εi j) isjust T2k(y) ⊗ 1/2, as in equation (4.18). Finally, (v) gives the projection for the two dimensionalcase, which can easily be seen as equivalent to the case in equation (4.18).

This gives a complete characterization of projections onto representations of irreversible spinfactors.

4.5. Projections onto reversible Jordan algebras

Let us now consider projections onto simple, reversible Jordan algebras. It turns out that again,we only have to consider two different cases: Either the Jordan algebra has a simple envelopingalgebra or an enveloping algebra that is the sum of two simple algebras. This section is in thespirit of [8] (which again is based on [35] and used heavily in [36] among others), i.e. the resultsare the same, but some of the proofs are different.To this end, consider a Jordan algebra of self adjoint elements Jh = span(A1, . . . , An), where thespan is meant to be as a real Jordan algebra (real, linear span + all Jordan products of Ai), denote by

80


R the real, associative algebra generated by products of the form∏m

i=1 Ai where Ai ∈ A1, . . . , An,i.e. the real algebra generated by the Jordan algebra. Then observe R + iR = S(J) by thedefinition of the enveloping algebra.

Lemma 4.20 ([33]). J is reversible if and only if Jh = Rh.

Proof. To shorten the notation at some places, we write∏1

i=n Ai := An . . . A1.Observe that Rh is reversible, as the set of self-adjoint matrices of an algebra is always reversible

since for(∏n

i=1 Aki +∏1

i=n Aki

)†=

∏ni=1 Aki +

∏1i=n Aki where the ki ∈ 1, . . . , n.

Now letJh be reversible, then by definition ofJh as real Jordan algebra generated by self-adjointelements, Jh ⊂ Rh.To see the other inclusion Rh ⊂ J , we consider the free associative real algebra generated byA1 . . . An, denoted by FA(A1, . . . , An). This algebra possesses a natural involution ∗ defined via(A1 . . . Am)† = Am . . . A1 and extended by linearity. We call an element A ∈ FA self-adjoint ifA†

= A. Considering the algebra of self-adjoint elements of FA, we call it FS, the free specialJordan algebra and note that it is generated as a Jordan algebra by Aii and reversible productsAk1 . . . Akm :=

∏ni=1 Aki +

∏1i=n Aki with the ki as above.

The universal property of free algebras ensures that there exists a unique homomorphism h :FA→ R(J) defined via Ai 7→ Ai. This then restricts to a Jordan-homomorphism of h : FS→ J .Both maps are, by construction, surjective. Furthermore, if † denotes also the complex conjugationon R(J), then h(A

†) = (h(A))† for all products A :=

∏mi=1 Aki with ki ∈ 1, . . . , n as before, hence

h(A†) = (h(A))† for all A ∈ R by linearity (R is a real algebra). Hence h is a ∗-homomorphism.

We want to show that for A ∈ Rh we have A ∈ J . Consider A ∈ Rh. If there exists an A ∈ FS s.th.h(A) = A, we are done. Hence suppose we only have an element A ∈ FA, s.th. h(A) = h(A†), butA† , A. To this end, split A into one part that is reversible and another part that is not:

A =∑

i1,...im

bi1,...,im(Ai1 . . . Aim + Aim . . . Ai1) +∑

i1,...im

ci1...,im Ai1 . . . Aim

where the coefficients are real. Now building the adjoint, we obtain:

A† =∑

i1,...im

bi1,...,im(Ai1 . . . Aim + Aim . . . Ai1) +∑

i1,...im

ci1...,im Aim . . . Ai1

This implies:

A − A† =∑

i1,...im

ci1...,im(Ai1 . . . Aim − Aim . . . Ai1)

Since by definition h(A − A†) = 0, we obtain via the linearity or h that

h

∑i1,...im

ci1...,im Ai1 . . . Aim

= h

∑i1,...im

ci1...,im Aim . . . Ai1

81


hence in particular

h

∑i1,...im

ci1...,im Ai1 . . . Aim

= h

12

∑i1,...im

ci1...,im(Aim . . . Ai1 + Ai1 . . . Aim)

which means that we have found an element A =∑

i1,...im(bi1,...,im + 12 ci1,...,im)(Ai1 . . . Aim +Aim . . . Ai1)

s.th. h(A) = A and A ∈ FS, thus A ∈ J , which was to be shown.

Lemma 4.21 ([8]). SetM := R ∩ iR, thenMh is a Jordan ideal of J .

Proof. As intersection of two linear subspaces,Mh ⊂ J is a real linear subspace. In addition,J ∗ R ⊂ R by the definition of R (and J ∗ Rh ⊂ Rh), i.e. M is a Jordan ideal.

Lemma 4.22. Let J ⊂ Md be a simple, unital reversible Jordan algebra such that its envelopingalgebra is given byMd. Then consider positive projections P :Md → J . There are two cases:

• M = J then P = 1 the identity, since the complex Jordan algebra generated by J isalreadyMd

• M = 0 then there exists an antiautomorphism α :Md →Md s.th. P = id +α2

Proof. This proof basically follows some arguments presented in [8] and [35].Since we have just seen thatMh is a Jordan ideal of J , which is a simple Jordan algebra, weimmediately haveMh = J orMh = 0. In the first case, any self-adjoint matrix A ∈ R is alsoan element of iR, hence R = S(J), which immediately implies thatMh is the set of self-adjointelements ofMd. But this algebra, as a real algebra, generatesMd as a complex Jordan-algebra,henceMd is fixed and the projection is necessarily the identity.LetMh = 0, then we can consider the direct sum R + iR. We first note that

R + iR =Md

which is true since J generatesMd by assumption (as a complex algebra) and every element inc ∈ Md can therefore be written as a sum of products with complex coefficients:

c =∑

i1,...,im

ci1,...,im Ai1 . . . Aim

=∑

i1,...,im

Re(ci1,...,im)Ai1 . . . Aim + i∑

i1,...,im

Im(ci1,...,im)Ai1 . . . Aim ∈ R + iR

Now consider the map α :Md →Md given by X + iY 7→ X† + iY† for X,Y ∈ R. First note thatthis map is an antiisomorphism:

α((X + iY)(W + iZ)) = α(XW − YZ + i(YW + XZ)) = W†X† − Z†Y† + iW†Y† + iZ†X†

= (W† + iZ†)(Y† + iX†)

82


for X,Y,W,Z ∈ R, hence α is an anti-automorphism, since it is obviously bijective and R-linearand

α(i(X + iY)) = iα(X + iY) ∀X,Y ∈ R

trivially. In addition, α(A†) = α(A)†, i.e. α is a Hermitian map. We also note that the fixed pointspace of α is precisely Rh + iRh (the complex Jordan algebra generated by Jh). ConsideringP := id +α

2 , we obtain that P2 = P as α α = id and hence P is a projection with P((Md)h) = Jh,since given A ∈ (Md)h, we have A = B + iC ∈ R + iR. Now A† = B† − iC†, hence B† = B andC† = −C in order for A† = A to hold, using the fact that the decomposition is a direct sum. Butthen α(A) = B† + iC† = B − iC and P(A) = B ∈ Jh. The last point to show is the positivity of P.However, this follows from the fact that α is positive, since given A = B + iC ∈ Md, A,C ∈ Rand A := B − iC ∈ Md, we obtain:

α(A†A) = α((B† − iC†)(B + iC)) = α(B†B + C†C − iC†B + iB†C)

= B†B + C†C − iB†C + iC†B = (B† + iC†)(B − iC) = A†A

Since a sum of positive maps is still positive, the proof is finished.

It is important to note that during the proof we saw a rather explicit construction of the antiauto-morphism α appearing in the lemma. In addition, we noted that the Jordan algebra consists of allthe elements that are fixed points of this antiautomorphism. This will be of value, when explicitlyconstructing the different projections. We also need uniqueness:

Lemma 4.23. Given a unital, reversible Jordan algebra J ⊂ Md such that S(J) =Md, there isa unique positive projection T onto J .

Proof. We cover the different cases of reversible embeddings with simple envelope separately:Case 1: Let J =Md(C), then the projection is the identity and uniqueness is trivial.Case 2: Let J be a reversible spin factor, then uniqueness is covered by lemma 4.8.For the other cases, we essentially follow the proof in [36], relying heavily on the appendix A.1.Let T be the projection ontoJ . Since T is linear, we can use the Peirce decomposition in equation(A.2). Let 1 =

∑ni=1 Pi be a decomposition into mutually orthogonal, minimal projections. Now

consider P(i j) := Pi + P j for i, j ∈ 1, . . . , n with i , j. Then for any A ∈ J we have

T (P(i j)AP(i j)) = T (2P(i j) ∗ (P(i j) ∗ A) − P2(i j) ∗ A)

(∗)= 2P(i j) ∗ (P(i j) ∗ T (A)) − P2 ∗ T (A) = P(i j)T (A)P(i j)

(4.20)

where we used lemma 2.25 in (∗) and the fact that P(i j) ∈ J . Given another positive, unitalprojection T , it suffices to show that T |P(i j)MdP(i j) = T |P(i j)MdP(i j) for all i, j, since we can decomposeany A ∈ Md into A =

∑i j λi jP(i j)MdP(i j) with some λi j (the decomposition is not unique, since

the P(i j) are not orthogonal).

83


Now, the Peirce decomposition tells us that P(i j)JP(i j) is a Jordan algebra, which means thatT |P(i j)JP(i j) is a positive, unital projection of P(i j)MdP(i j) onto P(i j)JP(i j). By construction,P(i j)JP(i j) is isomorphic to a unital Jordan algebra with a decomposition into two orthogonal,minimal projections: 1P(i j)JP(i j) = Pi + P j (by abuse of notation). Corollary A.11 implies thatP(i j)JP(i j) is therefore a spin factor. If we can show that S(P(i j)JP(i j)) = P(i j)MdP(i j), then weknow that T |P(i j)JP(i j) = T |P(i j)MdP(i j) since projections onto spin factors with simple envelope areunique (via lemma 4.8). To this end, we consider the different cases of reversible embeddingsof Jordan algebras with simple envelope that we have not yet covered and it clearly suffices toconsider one particular embedding, since all others are equivalent.Case 3: LetJ =Md(R)σ0 where σ0 is the inclusion induced embedding. The minimal projectionsare rank-one matrix units Eii for i = 1, . . . , d. These are obviously projections and they are inJ , hence, since they are rank one, they must be minimal. For i , j arbitrary, this means(Eii + E j j)(Md(R))σ0(Eii + E j j) (M2(R))σ0 , but S((M2(R))σ0) =M2(C), hence the envelopeis simple and the projections must be unique.Case 4: Let d be even and J =Md/2(H)σ0 where σ0 is the standard embedding of proposition3.6. A set of mutually orthogonal projections Pi ∈ J is given by Pi := Eii + Ed+i,d+i. In orderto see that these are also minimal projections, note that, using section A.1, we know that anynon-minimal projection P ∈ J contains a nonzero projection P ≤ P which is minimal. However,Pi has rank two, hence any minimal projection Pi ≤ Pi must have rank 1, i.e. Pi = Eii orPi = Ed+i,d+i. By the characterization of J in proposition 3.6, Pi < J , hence Pi was alreadyminimal. Then we see, similarly to case 3, Pi(Md/2(H))σ0 Pi (M2(H))σ0 , hence as in case 3,we obtain uniqueness, since the envelope is known to be the full matrix algebra.

Let us now turn to the case, where the enveloping algebra of our simple Jordan factor is a directsum of two simple algebras. Using [8, 34, 35], there is a general construction of such map in[36]. However, we will give simpler arguments along the lines of [36] that are not as general, buttailored to our needs. To avoid the abstract general methods of [36], we use our representationtheorem 3.3.From there we see two cases, where the enveloping algebra is a direct sum of two simple algebras:certain spin factors and universal embeddings of complex matrices.Since the spin factors were already covered, we only need to consider the case of Hermitianmatrices:

Lemma 4.24. By theorem 3.3, we know that every embedding J ofMd(C) is isomorphic to theuniversal embedding and therefore given by:

A 7→ U(A ⊗ 1m1 0

0 AT ⊗ 1m2

)U†

for all A ∈ Mdm1+dm2 and m1,m2 ∈ N and unitary U ∈ Mdm1+dm2 . Now consider any positive,unital T :Mdm1+dm2 →Mdm1+dm2 onto the Jordan algebra.

84


Let P1 be the projection onto the first direct sum factor (i.e. P1 :=(1dm1 0

0 0

)and P2 = 1 − P1).

Then T (P1) = λ1 with λ ∈ [0, 1]. Depending on λ, we have three possibilites:

• λ ∈ (0, 1), then

T (x) = (λTP1 + (1 − λ)TP2) Q(X) ∀X ∈ Mdm1+dm2

where Q : Mdm1+dm2 → Mdm1+dm2 is a projection onto S(J) = Md ⊗ 1m1 ⊕Md ⊗ 1m2

and for i = 1, 2

TPi(X) : S(J)→ J

X 7→ X

where X is defined such that XPi = XPi

• λ = 1, then

T (X) = TP1 QP1(X) ∀X ∈ Mdm1+dm2

where TP1 is defined as above and QP1 is a projection onto P1S(J)P1.

• λ = 0, then replace TP1 by TP2 and QP1 by QP2 with the same definitions.

Proof. Let us first give an overview over the proof. First, we make a few observations and generaldefinitions about the embedding in step 1. In the second step, we then see that T (Pi) ∝ 1 beforeusing the result to show that, if we restrict ourselves to the envelope of J , we can write downthe projections in step 3. Restricted to the envelope, the projections are a convex combinationof projections onto the subalgebras with simple envelope. In the fourth step, we then showthat elements in P1MdP2 + P2MdP1 vanish. Finally, in the last step, it will be shown that Tmust indeed be a concatenation of a projection onto the envelope and the restricted projectionsdiscovered in the third step. Step 1: First, we make a few observations about the Jordan algebra.With the definitions as in the lemma,Md ⊗ 1 ⊕ 0 = P1S(J)P1 = P1JP1 and likewise for P2. Inparticular, there exists an antiautomorphism:

χ : P1JP1 → P2JP2

χ

((A 00 0

)):=

(0 00 AT

)This is a very important observation, which we keep in mind for later.Step 2: Calculate T (P1).Obviously, 1 = T (1) = T (P1) + T (P2), hence if T (P1) = λ1, then T (P2) = (1 − λ)1. To this end,let 0 ≤ A ∈ J be arbitrary. Then

0 ≤ T (AP1) = T (A ∗ P1) = A ∗ T (P1)

85


by using that P1 commutes with the enveloping algebra of J and by using the equality conditionof the Jordan-Schwarz-inequality (see lemma 2.25). We have already encountered this scenarioin corollary 4.6, hence we obtain T (P1) = λ1 for some λ ∈ R. Since P1 ≥ 0 and P2 ≥ 0, alsoT (P1) ≥ 0 and T (P2) ≥ 0, which implies that λ ∈ [0, 1] and T (P2) = (1 − λ)1.The rest is now similar to the proof in [36] (proposition 6.4, 6.5), which is more general andneeds more technical results.Step 3: Let us first see how the projection must look like on the envelope of J : To this end,let X ∈ S(J), then we have, since PiS(J)Pi = PiJPi, that there exists a X ∈ J such thatXP1 = XP1 and a X ∈ J such that XP2 = XP2. Then

T (X) = T (X(P1 + P2)) = T (X ∗ P1) + T (X ∗ P2)

= T (X ∗ P1) + T (X ∗ P2)

= X ∗ T (P1) + X ∗ T (P2)

= λX + (1 − λ)X

= λT1(X) + (1 − λ)T2(X)

using again the equality conditions in the Jordan Schwarz-inequality (see lemma 2.25). In partic-ular, T1(X) = P1XP1 + χ(P1XP1) and also T2(X) = P2XP2 + χ−1(P2XP2).Step 4: To see what happens for arbitrary X ∈ Mdm1+dm2 , we will split the problem into two.First, we consider the off-diagonal elements X = P1XP2 + P2XP1 ∈ Mdm1+dm2 and in the finalstep, we consider elements X ∈ Mdm1 ⊕Mdm∈ . By linearity, this will complete the proof. First weclaim T (P1XP2 + P2XP1) = 0.We will again make use of the appendix A.1. Consider the standard universal embedding asshown in theorem 3.3. Let Ei j with i, j = 1, . . . , d denote the standard matrix units (see section3.1). Then Gi := Eii ⊗ 1m1 ⊕ Eii ⊗ 1m2 defines a set of minimal projections such that

∑di=1 Gi = 1.

Certainly, the Gi are projections and it is also clear (by construction) that Gi ∈ J since the Gi areof the form diag(A ⊗ 1m1 , A

T ⊗ 1m2) with A ∈ Md. Furthermore, they are necessarily minimal,since they are of minimal possible rank. In addition, using the multiplication table for matrixunits in equation (3.10), we find that they are orthogonal.Now, we consider the Peirce decomposition (equation (A.2)) of J with respect to this decompo-sition of the identity:

J =

d∑i=1

Gi

J d∑

i=1

Gi

=

d⊕i, j=1

Ji j

Since we know that P1 =∑d

i=1 Eii ⊗ 1m1 ⊕ 0 =:∑d

i=1 E(1)i (likewise for P2), we obtain that

T (P1XP2 + P2XP1) =

d∑i, j=1

T (E(1)i XE(2)

j + E(2)j XE(1)

i ) (4.21)

86


hence it suffices to show that each of the terms in the sum vanishes.To this end, define G(i j) := Gi + G j, then we find (the calculation being identical to equation(4.20)):

T(G(i j)XG(i j)

)= G(i j)T (X)G(i j)

for all X, hence the restriction of T to G(i j)Mdm1+dm2G(i j) is a positive, unital projections ontoa Jordan algebra. By construction, using lemma A.10, we have that the identity in G(i j)JG(i j)decomposes into two orthogonal minimal projections. In fact, one can easily see that G(i j)JG(i j)is isomorphic to the universal embedding of the spin factor V3, since we basically restrict to4 × 4-submatrices (modulo the ⊗1mi), which restricts from the universal embedding ofMd(C)to the universal embedding ofM2(C), which is the universal embedding of V3. However, westudied the universal embedding ofV2k+1 in corollary 4.11, where we found that the off-diagonalblocks vanished, i.e.

P(E(1)

i XE(2)j + E(2)

j XE(1)i

)= 0

where G(i j) = Gi + G j = (E(1)i + E(2)

i ) + (E(1)j + E(2)

j ). Since i, j were arbitrary, we obtain that eachof the terms in equation (4.21) vanishes. Hence the claim holds.Step 5: Now define projections Qi :Mdm1+dm2 →Mdm1+dm2 via:

Q1(X) = λ−1T (P1XP1)P1 ∀X ∈ Md

Q2(X) = (1 − λ)−1T (P2XP2)P2 ∀X ∈ Md

(if λ = 0 then set λ−1 = 0, else if λ = 1, set (1 − λ)−1 = 0. Then we do not need to distinguishbetween the cases). Then restricting Qi toMdmi , it is easy to see that Qi|Mdmi

:Mdmi →Mdmi

is a positive, unital projection onto PiJPi. Positivity follows since P1,2 ≥ 0 and maps of theform P(·)P for arbitrary P ≥ 0 are completely positive. Unitality follows, since Qi|Mdmi

(1) =

Qi(Pi)|Mdmi= 1Mdmi

. Finally, it is a projection, because T is a projection. This implies that therange of Q1,2 is as stated, and (consider only Q1, Q2 is analogous) for A ∈ P1JP1 we haveQ1(Q1(A)) = λ−2T (P1(T (P1AP1)P1P1))P1 = λ−2T (T (P1AP1) ∗ T (P1))P1 = λ−1T (P1AP1)P1similar to the calculation below.SinceMdmi = PiJPi⊗Mmi , we also know that Qi(Y⊗Z) = Y⊗1 tr(Zρi) for some state ρi ∈ Mmi

and all Y ⊗ 1 ∈ PiJPi and Z ∈ Mmi . This follows from lemma 4.5.On the other hand, we can define:

Q(X) := Q1(X) + Q2(X)

and obtain a projection Q :Mdm1+dm2 →Mdm1+dm2 onto S(J). For arbitrary X ∈ Mdm1+dm2 wenow have:

T (Q(X)) = T (λ−1T (P1XP1)P1 + (1 − λ)−1T (P2XP2)P2)

87

Martin Idel Construction of positive projections onto reversible Jordan algebras

= λ−1T (T (P1XP1) ∗ P1) + (1 − λ)−1T (T (P2XP2) ∗ P2)

= T (P1XP1) ∗ λ−1T (P1) + T (P2XP2) ∗ (1 − λ)−1T (P2)

= T (P1XP1) + T (P2XP2) = T (X)

where we made use of the equality condition in the Jordan Schwarz-inequality (lemma 2.25)again.Hence we can replace T by T |S(J) Q and as stated above,

T |S(J) = λ(P1XP1 + χ(X)) + (1 − λ)(P2XP2 + χ−1(X))

which completes our proof.

4.6. Construction of positive projections onto reversible Jordanalgebras

We can now collect our knowledge about representation theory and projections from the previouschapters and sections to give precise constructions of positive projections onto reversible subalge-bras. We need these to construct and characterize the fixed point spaces of trace-preserving maps.Using theorem 3.3, we decompose the problem into five different paragraphs:

C* algebras

This case is already known and studied in e.g. [40] (chapter 1). We have everything in place toprove it again.

Proposition 4.25 ([40], Prop. 1.5). Let K ∈ N the Jordan algebra

J := U

K⊕i=1

Mdi ⊗ 1mi

U† ⊂ Md (4.22)

DenotingHk and its isometries Vk and partial traces trk,i as obtained from equation (4.11). Then,any unital, positive, linear projection T ∗ ofMd onto J has the form:

T ∗(A) =

K∑k=1

Vk(trk,2

[(V†k AVk)(1dk ⊗ ρk)

]⊗ 1mk

)V†k (4.23)

where ρk ∈ Mmi is a state.

Proof. Using equation (4.2), we can see that T is a sum of projections Tk, where Tk are projectionsonto the individual summands of the direct sum. Now via lemma 4.5, these maps are given asstated in the proposition.

88


Corollary 4.26. Consider the scenario of the proposition. Given a positive, trace-preservingmap T with a full-rank fixed point such that the fixed point algebra of T ∗ is given by J , then itsfixed point projection T∞ has the form:

T∞(A) =

K∑k=1

Vk(trk,2(V†k AVk) ⊗ ρk

)V†k

where ρk ∈ Mmk > 0 is a state.

Proof. For arbitrary A,C ∈ Md, using the form of T ∗∞ as in the proposition:

tr

C† K∑k=1

Vk(trk,2


]⊗ 1mk

)V†k

=

K∑k=1

tr((V†k CVk)†

(trk,2


]⊗ 1mk

))=

K∑k=1

tr((V†k CVk)†

(trk,2


]⊗ 1mk

))(∗)=

K∑k=1

tr(trk,2(V†k CVk)† ⊗ ρk(V†k AVk))

=

K∑k=1

tr((Vk[trk,2(V†k CVk)† ⊗ ρk]V†k )†A)

To justify (∗), use equation (4.13) and set C, E = 1 and D = ρk (obviously, the subscript k for thedifferent Hilbert spaces does not alter anything in the equation).Finally, we need ρk to be positive definite, since otherwise, T has not a full-rank fixed point.

Symmetric Jordan algebras

For K ∈ N consider the Jordan algebra

J := U

K⊕i=1

JTdi⊗ 1mi

U† ⊂ Md (4.24)

where JTdi

= S i(Mdi(R)σ0 + iMdi(R)σ0)S †i with the universal embedding σ0 induced by inclusionand unitary S i ∈ Mdi . As before, we can push the S i into the unitary U to obtain a differentunitary U.

Proposition 4.27. Let J be as in equation (4.24) and defineHk and its isometries Vk and partialtraces tri,k as before in equation (4.11). We obtain that any unital, positive, linear projection T ∗

89


ofMd onto J has the form:

T ∗(A) =

K∑i=1

Vk

(12

(trk,2

[V†k AVk(1dk ⊗ ρk)

]+ trk,2


]T)⊗ 1mk

)V†k (4.25)

where ρk ∈ Mmi is a state.

Proof. First note that the enveloping algebra of any summand JTdi

is isomorphic to a simplematrix algebra. In the same way as in the proof of proposition 4.25, lemma 4.4 and 4.5 show thatT is a sum of concatenated projections:

MdT (i)

1−→Mdimi

T (i)2−→Mdi ⊗ 1mi

T (i)3−→ JT

di⊗ 1mi

T (i)1 is given by lemma 4.4, T (i)

2 is given by lemma 4.5. Moreover, lemma 4.22 tells us thatT (i)

3 = id+α2 ⊗ 1mi , where α is an antiisomorphism, s.th. J consists of the elements ofMdk which

are invariant under α. But then α = θ, the transposition, for the inclusion-induced embedding.We then have T =

∑k T (k)

3 T (k)2 T (k)

1 . Knowing that T (k)2 T (k)

1 has the form of the map inproposition 4.25, hence we obtain:

T ∗(A) =

K∑i=1

Vk

trk,2[V†k AVk(1dk ⊗ ρk)

]+ trk,2


]T

2⊗ 1mk

V†k

which is exactly the stated result, where we used the linearity of the tensor product. In principle,one could put the transpose inside the partial trace, since the trace is invariant under transposition.

Corollary 4.28. Consider the situation as in proposition 4.27. Given a positive, trace-preservingmap T with a full-rank fixed point such that the fixed point algebra of T ∗ is given by J , then itsfixed point projection T∞ has the form:

T∞(A) =

K∑k=1

Vk

trk,2

V†k AVk + (V†k AVk)T

2

⊗ ρk

V†k


Proof. Using linearity, the proof is analogous to the proof of the corollary for the C*-algebracase (corollary 4.26), where we also have to use that the transposition (in the computational basis)is self-adjoint. The fact that ρk must be positive definite follows from the fact that the C*-algebramust contain a full-rank fixed point.

90


Quaternionic Jordan algebrasFor K ∈ N consider the Jordan algebra

J := U

K⊕i=1

JHdi⊗ 1mi

U† ⊂ Md (4.26)

where di is even and JHdi⊂ Mdi and JH

di= S (Mdi/2(H)σ + iMdi/2(H)σ)S ∗ with the universal

embedding σ0 as in proposition 3.6, and unitary S ∈ Mdi . As before, we can push the unital Sinto U to obtain U.

Proposition 4.29. Let J be a Jordan algebra as in equation (4.26) and define Hk and itsisometries Vk and partial traces trk,i as from equation (4.11), then we have: Any unital, positive,linear projection T ∗ ofMd onto J has the form:

T ∗(A) =

K∑i=1

Vk

trk,2[V†k AVk(1dk ⊗ ρk)

]+ Q−1 trk,2


]TQ

2⊗ 1mk

V†k (4.27)

where ρk ∈ Mmi is a state and we recall Q =

(0 1

−1 0

)in the definition of the universal embedding.

Proof. First note that the enveloping algebra of any summand JHdi

is isomorphic to a simplematrix algebra. In the same way as in the proof of proposition 4.27, we can see that T is a sum ofmaps that are a concatenation of projections Tk : S(JH

dk)→ JH

dkand projections onto the envelope.

Moreover, again via lemma 4.22, we know that Tk = 1+α2 , where α is an antiisomorphism, s.th.

J consists of the elements ofM2dk which are invariant under α. But then α = Q−1(.)T Q with Qspecified in proposition 3.6, which leads to the stated result.

Corollary 4.30. Consider the situation as in proposition 4.29. Given a positive, trace-preservingmap T with a full-rank fixed point such that the fixed point algebra of T ∗ is given by J , then itsfixed point projection T∞ has the form:

T∞(A) =

K∑k=1

Vk

trk,2

V†k AVk + (Q ⊗ 1mk )−1(V†k AVk)T (Q ⊗ 1mk )

2

⊗ ρk

V†k


Proof. Using linearity, the proof is analogous to the proof of the corollary for the C*-algebracase (corollary 4.26), where we also have to use that the transposition (in the computational basis)is self-adjoint. In addition, we use

Q−1 trk2(.)Q ⊗ 1mk = (Q ⊗ 1mk )−1(trk2(.) ⊗ 1mk )(Q ⊗ 1mk )

as well as QT = −Q, Q−1 = Q and the cyclicity of the trace. The fact that ρk must be positivedefinite follows from the fact that the C*-algebra must contain a full-rank fixed point.

91


Complex Hermitian algebrasFor K ∈ N consider the Jordan algebra

J := U

K⊕i=1

JCdi

U† ⊂ Md

where JCd = S diag(a, aT )S †, where S ∈ Mdim

(1)i +dim

(2)i

(C) is a unitary and the envelope isisomorphic to the universal envelope and a = a ⊗ 1m(1)

i⊂ Mdim

(1)i

and a = a ⊗ 1m(2)i⊂ Mdim

(2)i

forall a inMdi(C).We again shift the unitary S outside of the direct sum and construct another unitary U.Denote by Pi the projection onto the i-th summand of J and by P(1)

i (P(2)i ) the projection onto

the first (second) factor of the i-th envelope.The direct sum structure of J leads to a decomposition of the Hilbert space J acts on in the sameway as in equation (4.15). We denote the isometries Vi, V ( j)

i as in the case following equation(4.15) and check that our definition of P( j)

i coincides.

Proposition 4.31. Consider the scenario described in this paragraph. Given a unital, positive,linear projection T ∗ of Md onto J , there are states ρ(1)

i ∈ Mm(1)i

and ρ(2)i ∈ Mm(2)

i, there are

λ(1)i , λ(2)

i ∈ [0, 1] with λ(1)i + λ(2)

i = 1, such that

T †(A) =

K∑i=1

2∑o,p=1

λ(o)i V (p)

i

(tr(o)

i,2

[V (o)∗

i AV (o)i

(1di ⊗ ρ

(o)i

)]⊗ 1m(p)

i

)δopTV (p)∗

i (4.28)

A different, less explicit but more accessible form of these projections was given in lemma 4.24and we define δopT to mean that we take the transposition if o , p and if o = p we do not.

Proof. This follows from lemma 4.24 above. We have just written out the involved maps.The isometries V (o)

i provide the restrictions that are provided by the Po in the lemma, the parttr(o)

i,2

[(V (o)∗

i AV (o)i )(1di ⊗ ρ

(o)i )

]⊗ 1m(p)

iis, as above, the projection ontoMdk ⊗Mm(o)

i. In the given

embedding, the isomorphism χ in lemma 4.24 is nothing else but the transposition and switching

of a matrix block (we have χ :(A 00 0

)7→

(0 00 AT

)), this is reflected by the two summands in

the inner sum (the inverse χ−1 looks the same) in the fact that if o , p, then we transpose themap.

Corollary 4.32. Consider the scenario described in this paragraph. Given a positive, trace-preserving map T with a full-rank fixed point such that the fixed point algebra of T ∗ is given byJ , then its fixed point projection T∞ has the form:

T∞(A) =

K∑k=1

2∑o,p=1

λ(p)i V (p)

i

(tr(o)

i,2

[V (o)∗

i AV (o)i

]⊗ ρ

(p)i

)δopTV (p)∗

i (4.29)

92


with λ(1)i , λ(2)

i ∈ (0, 1) and ρ(o)i > 0 states as in proposition 4.31.

Proof. Using linearity, the proof is analogous to the proof of the corollary for the C*-algebra case(corollary 4.26), where we also have to use that the transposition (in the computational basis) isself-adjoint. The fact that ρ(o)

i must be positive definite follows from the fact that the C*-algebramust contain a full-rank fixed point and similarly, it follows that λ(o)

i , 0 (see also the later proofof corollary 4.11).

Reversible spin factorsSince we have already covered the spin factors, there is nothing left to do. We just remindthe reader that certain spin factors are reversible. For these cases (namelyV2,V3 and certainembeddings ofV5), we have also the reversible characterization, which might be nicer than thecharacterization we gave in section 4.4.

Final remarks about decomposabilityThe explicit construction of the projections lets us also easily characterize them in the followingsense (see also theorem 5.1):

Proposition 4.33. A unital, positive projection onto a reversible Jordan algebra is decomposable.

Proof. We will give a different proof in proposition 5.5. Here, the theorem follows by inspection:By the preceding propositions of this section, every positive, unital projection onto a reversibleJordan algebra consists of a sum of maps, each of which contains maps of the form T (X) = S XS †

with unital S , which are completely positive, it contains linear functionals (projections onto theenvelope tensored with some identity), which are also completely positive maps and finally, theonly non-completely positive part is the transposition. Hence the maps are decomposable.

93

5. Characterizations of positive projectionsonto fixed point algebras

In chapter 4, we have constructed all fixed point projections resulting from positive maps with afull-rank fixed point. In passing, we have also seen that the projections onto reversible Jordanalgebras are decomposable. Here, we want to elaborate on this point and further analyze positivemaps. In section 5.1, we obtain that in fact, all projections are onto reversible representations aredecomposable. Furthermore, in section 5.2, we obtain that the representations onto spin factorsare atomic, i.e. indecomposable in the strongest sense. More precisely, we obtain the followingresults:

Theorem 5.1 (Decomposability of fixed point projections). Given a trace-preserving, positive,linear map T :Md(C)→Md(C), we obtain for its unital adjoint T ∗:

• T is decomposable iff T ∗ is decomposable and if T ∗ is decomposable, then T ∗∞ is decom-posable, thus also T∞ is decomposable.

• If T ∗∞ is decomposable, then T is not necessarily decomposable.

Now consider only the case where T ∗∞ projects to a simple, unital Jordan algebra J , then wehave:

• T ∗∞ is decomposable, iff J is reversible.

• In particular, if T ∗∞ is indecomposable, then J must be a spin factor and T ∗∞ is even atomic.

The main value of this theorem is the statement that these projections are not necessarily decom-posable and is essentially a combination of [8], [36] and [38]. These sources deal with positiveprojections onto Jordan algebras in the more general setting of a C*-algebra, instead of a finitedimensional matrix algebra, but we will break down their proofs to the finite dimensional case. Inessence, we need to combine our knowledge of how projections onto reversible factors look like.If we recall our theorem on representations of Jordan algebras (thm. 3.3), we can find that thedimensions of the summands of Jordan algebras, where nondecomposable maps can be found,are restricted to powers of 2, since we can only find spin factors in these dimensions.The distinction between decomposability and indecomposability combines and simplifies resultsfrom [8, 11, 12, 15, 33, 34, 35, 36] to finite dimensions, while the statement that indecomposableprojections onto spin factors are even atomic apparently first appeared in [28] (another proof isgiven in [10]), we will however use [38] as a reference.

94

Martin Idel Decomposability of unital projections onto Jordan algebras

5.1. Decomposability of unital projections onto Jordan algebras

Working towards the proof of theorem 5.1, we first need some basic facts about decomposability.The first two lemmata appear in the literature in several places (mentioned e.g. in [40]).

Lemma 5.2. If T is a positive, linear, decomposable map, then so is T ∗.

Lemma 5.3. If T is a positive, linear, unital map, which is decomposable, then T∞ is alsodecomposable.

Proof. We have seen above that T∞ = limN→∞

1N

∑Nn=1 T n. The sum of two decomposable maps is

clearly again decomposable, since any positive linear combination of completely positive mapsis again completely positive (via Choi’s isomorphism, completely positive maps correspondto positive semidefinite matrices and a sum of positive semidefinite matrices is still positivesemidefinite).In addition, the composition of decomposable maps is decomposable, since for T := T1 + θ T2and S := S 1 + θ S 2 with completely positive T1,T2, S 1, S 2 we have:

T S = T1 T2 + θ T2 θ S 2 + T1 θ S 2 + θ T1 S 2 (5.1)

Now using that T is completely positive if and only if θ T θ is completely positive (via Choi’sisomorphism: Given a maximally entangled state |Ω〉 in the basis of the transposition, we have(T ⊗ 1)(|Ω〉〈Ω|) = (1 ⊗ θT ∗θ)(|Ω〉〈Ω|)), we obtain that the first two maps in this decompositionare cp, while the last two are co-cp. In addition, from positivity, we obtain 0 ≤ T1 ≤ T and0 ≤ T2 ≤ T .Using these facts, we immediately see that every term in the limit leading to T∞ is decomposable,hence also the limit must be decomposable (where we used that the limit of a positive map isagain positive).

Another question that might be asked is, whether the converse holds, i.e. whether decomposabilityof T∞ implies decomposability of any T that leads to T∞. This is not true, since the followingcounterexample holds.

Proposition 5.4. Consider the Breuer-Hall map T BH :M4 →M4 with T BH(ρ) = 12 (tr(ρ)1− ρ−

UρT U∗) with the antiunitary

U :=(iσ2 00 iσ2

), σ2 =

(0 −ii 0

)then T BH

∞ (X) = tr(X)1, which is obviously a completely positive map, hence T∞ is triviallydecomposable, but T is indecomposable.

95

Martin Idel Decomposability of unital projections onto Jordan algebras

Proof. We first note that U is really antiunitary, via: U = i diag(σ2, σ2), then it holds thatU∗ = −i diag((σ2)∗, (σ2)∗) = −i diag(σ2, σ2) = −U. With this knowledge, it was proven in [11]that T BH is positive and indecomposable. Obviously, by the unitarity of U, we have that T BH isdoubly stochastic:

T BH(1) =12

(tr(1)1 − 1 − U1T U∗) = 1

tr(T BH(ρ)) =12

(tr(ρ) tr(1) − tr(ρ) − tr(UρT U∗)) = tr(ρ)

hence it is in particular of the type we analyze (positive, unital map with full-rank fixed-point). Inorder to calculate T BH

∞ , we consider T BH(X) of an arbitrary matrix

X =

a b c de f g hi j k lm n o p

⇒ T BH(X) =

12

p + k 0 −n − c j − d

0 p + k m − g −h − i−n − i d − j a + f 0g − m −n − c 0 a + f

From this calculation, we immediately obtain that any fixed point X ∈ FT fulfills a = f = k = pand b = e = l = o = 0. Also, all the other entries must be zero, via:

m =g − m

2∧ g =

m − g2

⇒ m =g3∧ g =

m3

⇒ m = g = 0

h =−h − i

2∧ i =

−h − i2

⇒ h = i = −i = −h ⇒ h = i = 0

The cases j = d = 0 and n = c = 0 work the same way. Hence the only fixed points are multiplesof the identity. But then, T BH

∞ (ρ) = tr(ρ)1, which is a completely positive map, hence triviallydecomposable.

Let us give the first characterization of decomposability, which will be extended in the nextsection:

Proposition 5.5. Let J ⊂ Md(C) be a finite dimensional, unital Jordan algebra and T :Md →

Md be a positive, linear projection, such that T (Md(C)) = J . Then T is decomposable if andonly if J is reversible.

Proof. As mentioned above, this theorem generalizes to arbitrary dimensions, we give a shorterproof for finite dimension, which relies heavily on the proof for arbitrary dimensions.The proof that if Jh is irreversible, then T is not decomposable is a direct corollary of proposition

96

Martin Idel Properties of indecomposable projections onto spin factors

5.6 and we will postpone the proof. The converse direction was already noted in proposition 4.33.For the reader’s convenience, we will give another proof in the spirit of [36], making use not ofthe explicit form of the projections but our more abstract knowledge:Given a reversible algebra J , we can immediately restrict to the case where J is simple. Hencewe have two cases to consider: Either the enveloping algebra is irreducible or it is the direct sum oftwo irreducible factors. In principle, we have characterized all projections onto reversible Jordanalgebras in section 4.6 and from there, we can read off that all projections are decomposable (seeproposition 4.33). For the reader’s convenience, we also give a slightly different proof in thespirit of [36]:First consider the case of simple enveloping algebra. By lemma 4.22, T is a concatenation of amap onto the enveloping algebra, which is a completely positive map and a projection from S(J)onto J . Then by lemma 4.22, the projection is either T = id, which is obviously completelypositive or T = id +α

2 with an antiisomorphism α. However, an antiisomorphism of a full matrixalgebra is inner (see Skolem-Noether, theorem 3.15), hence completely copositive, which meansthat the map is decomposable.Now consider the case, where the envelope is a direct sum of two simple matrix algebras. Wehave seen that in all possible cases, T can be written as T = T |S(J) Q with Q a projectiononto S(J). Since projections onto C*-algebras are decomposable (by lemmata 4.4 and 4.5), Qis decomposable. Furthermore, we have an explicit construction of T |S(J) in terms of a sum oftwo maps, where one is of the form P.P for positive P and the other is of the form χ(P.P) forpositive P and an antiautomorphism χ. Since antiautomorphism are completely copositive (infact, χ turned out to be mainly the transposition, which is obviously completely copositive) andsums of decomposable maps are decomposable, the projections are decomposable.We will not prove the converse direction, since we give a stronger statement in the next section,which implies this result. A direct proof can be found in [36], which uses the fact that thereis a strong connection between decomposable maps and Jordan homomorphisms in the sensethat any decomposable map can be written in the form T (X) = V†π(X)V with isometries V andJordan homomorphisms π ([37]) and also, any Jordan *-homomorphism T between a C*-algebraand an algebra of bounded operators on a separable Hilbert space can be written as a sum of a*-homomorphism and a *-antihomomorphism ([24, 33]). A short survey of these and similarproperties can also be found in the appendix of [24]. However, the proof in the next section seemsto better highlight the intuition behind why we obtain the result.

5.2. Properties of indecomposable projections onto spin factors

Now we want to show that, even stronger, all indecomposable projections are not even decompos-able in any weaker sense.

Proposition 5.6. Let T : Md → Md be a positive, unital projection onto an irreversible spinfactor. Then T is atomic.

97


Proof. We follow along the lines of the proof of theorem 2.3.4. in [38].First recall proposition 3.4. There, we have seen that spin factors Vk have only irreversiblerepresentations for k ≥ 6, k = 4 and some irreversible representations for k = 5. The crucialobservations of the proofs was always that given S 1, S 2, S 3, S 4 ∈ Vk part of the underlying spinsystem, then S := S 1S 2S 3S 4 = 1

2 (S 1S 2S 3S 4 + S 4S 3S 2S 1) cannot belong to Vk. This will bethe key observation, namely, if T were 2-decomposable, then S ∈ Vk.Let Vk be the irreducible spin factor that is the image of T , a positive, unital map. The spinsystem V4 is a subset of the spin system Vk defining Vk, hence V4 is a subalgebra of Vk. Weknow thatV4 is unique up to isomorphism of the envelope and that its envelope is isomorphictoM2 ⊗M2. Recall furthermore, that the universal embedding is given by S 1 = σ3 ⊗ 1, S 2 =

σ1 ⊗ 1, S 3 = σ2 ⊗ σ3. Let θ denote the transposition onM2. Define the map:

β :M2 →M2

σi 7→ σ2σiσ2

hence we have β(σ1) = −σ1, β(σ2) = σ2, and β(σ3) = −σ3. Then,

α := θ ⊗ (θ β)

is an anti-automorphism (β is an automorphism and the transposition is an anti-automorphism).Since σT

2 = −σ2 and the other σi are symmetric, we obtain that α(A) = A for all A ∈ V4.Now S(V4) ⊂ Md is an embedded C*-algebra, hence (using proposition 4.25) we have acompletely positive, unital projection P1 :Md →Md onto S(V4) and Q := αP1 is a projectiononto V4 such that Q θ′ is 2-positive, where θ′ is the extension of the transposition θ onM2 toMd (which, in the standard basis, corresponds just to the normal transposition). Furthermore,Q(A) = A for all A ∈ V4.Suppose T were 2-decomposable, i.e. T = T1 + T2 such that T2 = T ′2 θ and T1,T ′2 are 2-positive.Then define for ε > 0,

Tε := (1 − 2ε)T + ε id +εQ (5.2)

The crucial idea of the proof is to show that for arbitrary ε > 0 small enough, if T is 2-decomposable, we have that Tε(S ) = S , which would imply that

T (S ) = limε→0

Tε(S ) = S (5.3)

hence S ∈ Vk, which we know to be impossible, i.e. we have a contradiction.To this end, decompose Tε = T1,ε + T2,ε with T1,ε := (1 − 2ε)T1 + ε id is 2-positive andT2,ε := (1 − 2ε)T2 + εQ is 2-copositive. Moreover, since S i are invertible Q(1) is invertibleand therefore, H := T1,ε(1)

12 and K := T2,ε(1)

12 are invertible for every ε > 0 small enough.

Therefore,

T1,ε := H−1T1,εH−1

98


T2,ε := K−1T2,εK−1

define unital, positive maps. Also, since H ≥ 0, K ≥ 0, T1,ε (T2,ε) is a concatenation of acompletely positive with a 2-positive (2-copositive) map and therefore 2-positive (2-copositive)and we have:

Tε(A) = HT1,εH + KT2,εK

Since T (A) = A for all A ∈ Vk and Q(A) = A for all A ∈ V4, we have Tε(A) = A for all A ∈ V4.Let A ≥ 0 and A ∈ V4, then Tε(A) = A implies T1,ε(A) ≤ A and T2,ε(A) ≤ A by the positivity ofthe two maps. Now consider a projection E ∈ V4, then we have 0 ≤ Ti,ε(E) ≤ E and hence weobtain (1 − E)Ti,ε(E) = 0 = ETi,ε(1 − E) for i ∈ 1, 2. But then,

0 = (1 − E)Ti,ε(E) − ETi,ε(1 − E) = Ti,ε(E) + ETi,ε(1)

hence Ti,ε(E) = ETi,ε(1) = Ti,ε(1). Since the spectral projections of J span J (see appendixA.1), it follows that Ti,ε(A) = ATi,ε(1) for all A ∈ A, a result which is of course trivial forprojections onto unital Jordan algebras.In our case this implies

T1,ε(A) = H2A = AH2

T2,ε(A) = K2A = AK2

and not only Tε(A) = A. In particular, HA = AH and KA = AK for all A ∈ V4 and thus,

Ti,ε(A) = A

for all A ∈ V4 and i ∈ 1, 2. Since Ti,ε is positive and unital, Ti,ε(S i) = S i for all i = 0, . . . , 4.Since T1,ε is 2-positive, it is a Schwarz map by lemma 2.23. One can easily see that, given aSchwarz-map φ :Md →Md, if φ(A†A) = φ(A†)φ(A) for some A, then φ(A†B) = φ(A†)φ(B) forall B ∈ Md. The proof follows from the proof of lemma 2.25 by replacing the Jordan product withthe matrix product and using the Schwarz-inequality instead of the Jordan-Schwarz-inequality.Hence we obtain:

T1,ε(S ) = T1,ε(S 1)T1,ε(S 2S 3S 4)

= . . .

= T1,ε(S 1) . . . T1,ε(. . . S 4) = S 1 . . . S 4 = S

Likewise, T1,ε(S ) = S using the anti-Schwarz inequality and in addtion the fact that S 1S 2S 3S 4 =

S 4S 3S 2S 1. Then S is a fixed point of Ti,ε. Since we have seen above that HA = AH andKA = AK for all A ∈ V4, HS = S H and KS = S K and therefore

Tε(S ) = HT1,ε(S )H + KT2,ε(S )K = HS H + KS K = (H2 + K2)S = (Tε(1))S = S

by the unitality of Tε. But then S is a fixed point of Tε for all ε > 0 small enough, hence S ∈ Vk,which was proven to be impossible. Therefore, T cannot be 2-decomposable.

There are attempts to characterize atomic maps even finer. One attempt is [37].

99

6. Fixed point spaces of positive,trace-preserving maps and the peripheralspectrum

Now, we are finally in a position to extend the results presented in [40] (sections 6.4 and 6.5)to general positive maps. Since we have constructed all positive projections, this is but a meretechnical issue. In addition, we will also study the peripheral spectrum of such maps.

6.1. Fixed point spaces of positive, trace-preserving maps

In chapter 4, we have constructed all projections onto finite dimensional, unital Jordan algebras.We have also constructed their adjoints. Now given a positive, trace-preserving map T with afull-rank fixed point, then T∞ = (T ∗∞)∗ via theorem 5.1 and T ∗∞ is a projection onto a unital Jordanalgebra. Looking at the adjoints of the projections onto Jordan algebra, their images classify thepossible fixed-point spaces of T .

Theorem 6.1 (Fixed point spaces of positive, trace-preserving maps). Let T :Md →Md be apositive, trace-preserving map. Then the fixed point algebra FT has the following structure:

FT = U

0 ⊕ K1⊕

i=1

Mdi ⊗ ρi

⊕ K2⊕

i=1

JCdi

⊕ K3⊕

i=1

JTd′i⊗ ρ′i

⊕

K4⊕i=1

JHd′′i⊗ ρ′′i

⊕ K5⊕

i=1

Jspinfullki⊗ ρ′′′i

⊕ K6⊕

i=1

Jspinsumki

U∗

(6.1)

where U ∈ Md is a unitary and 0 < ρi, ρ′i , ρ′′i , ρ

′′′i are states and the unknown factors are given

by

1. d′i ∈ N, JTd′i

= (Md′i (R) + iMd′i (R)).

2. d′′i ∈ 2N, d′′i ≥ 6, and set d′′i = 2ki, then JHd′′i⊂ Md′′i and JH

d′′i= (Mki(H)σ + iMki(H)σ) is

the complex extension of the embedding σ of the Jordan algebra of Hermitian quaternionsin proposition 3.6.

3. di ∈ N, di ≥ 3, JCdi

= diag(λiA, (1 − λi)AT ) with A = A ⊗ ρi ⊂ Mdimi , A = A ⊗ ρi ⊂ Mdimi

for all A ∈ Mdi(C) and some states 0 < ρi ∈ Mmi , 0 < ρi ∈ Mmi for some mi, mi ∈ N.

100

Martin Idel Fixed point spaces of positive, trace-preserving maps

4. Jspinfullki

with 4 ≤ ki = 2ni or 4 ≤ ki = 2ni + 1 is a spin factor of dimension ki + 1 andembedding given in proposition 3.7 for ki even and proposition 3.8 for ki odd.

5. Jspinsumki

with 4 ≤ ki = 2ni + 1 odd is a spin factor of dimension ki + 1 and given S j,j = 0, . . . , 2ni + 1 an embedding of the spin-system intoM2ni as in proposition 3.8, thenJspinsum

kiis spanned by elements of the form diag(λiS j ⊗ ρ

(1)i ,±(1−λi)S j ⊗ ρ

(2)i ) with + for

j = 0, . . . , 2ni and − for j = 2ni + 1, 0 < ρ(1)i ∈ Mm(1)

iand 0 < ρ(2)

i ∈ Mdm(2)i

some states,λ ∈ (0, 1).

The decomposition given here is unique up to isomorphism.In particular, we have that the maximum-rank fixed point ρ is given by ρ := T∞(1), i.e.

ρ = U

0 ⊕ K1⊕

i=1

1di ⊗ ρi

⊕ K2⊕

i=1

1di ⊗ λiρi ⊕ 1di ⊗ (1 − λi)ρi

⊕ K3⊕

i=1

1di ⊗ ρ′i

⊕

K4⊕i=1

1di ⊗ ρ′′i

K5⊕

i=1

12ni ⊗ ρ′′′i

K6⊕

i=1

12ni ⊗ λiρ(1)i ⊕ 12(ni) ⊗ (1 − λi)ρ

(2)i

U†

(6.2)

and ρ−1/2FTρ−1/2 is a semi-simple Jordan algebra, which is unital iff ρ is full-rank. Thus the

space of fixed points FT is a Jordan algebra under the modified Jordan-product:

A ∗ B :=12

(T∞(1)−1/2AT∞(1)−1BT∞(1)−1/2 + T∞(1)−1/2BT∞(1)−1AT∞(1)−1/2) (6.3)

for A, B ∈ FT .

Let us note the following direct corollary:

Corollary 6.2. Let T be a positive map, then it is doubly-stochastic iff its fixed point algebra is aunital Jordan algebra.

Proof. This follows directly from the fact that T doubly stochastic, then T∞ doubly stochastic bythe definition of the fixed point projection.

Now let us comment on the theorem: Assume that T had a full-rank fixed point, then the firstsummand in equation (6.1) is the classical case for completely positive maps (see e.g. [40]). Thetheorem is not very surprising, if we recall that in theorem 4.1, we have seen that the projectionsonto Jordan algebras with simple envelope are unique and self-adjoint and all projections areconcatenation of projections onto the C*-envelope and projections from the envelope onto theJordan algebra. In addition, the terms arising from Jordan algebras with nonsimple envelopenecessarily need to be of the form diag(λA, (1 − λ)AT ), since they must be trace-preserving, as Tis trace-preserving. The intuitive reasoning is that a map A 7→ diag(A, AT ) for all A ∈ Md as itoccurs in the projection onto a unital Jordan algebra is not trace-preserving. An obvious way tomake it trace-preserving and positive is to set A 7→ diag(λA, (1 − λ)AT ) for some λ ∈ [0, 1]. Thisis what happens here.Let us now proceed to a formal proof of theorem 6.1:

101

Martin Idel Fixed point spaces of positive, trace-preserving maps

Proof. The proof is along the lines of the proof for completely positive maps in [40] (theorem6.14ff).First assume T has a full-rank fixed point, then the fixed point space of T ∗ is a unital Jordanalgebra by theorem 6.1. The proof then follows from inspection of T∞. We know that T ∗ is apositive, unital projection onto a Jordan algebra by theorem 2.26. Hence T ∗ is of the form ofprojections discussed in theorem 4.1. Recall that we already calculated the adjoint maps T∞. Forthe reader’s convenience, we recap the results here:Consider the decomposition of the underlying Hilbert space in equations (4.11) and (4.15) togetherwith the corresponding isometries defined there. Then we obtain:

• T ∗ is a projection onto the first summand in the theorem (C*-algebras), then by corollary4.26:

T∞(A) =

K1∑k=1

Vk(trk,2(V†k AVk) ⊗ ρk

)V†k

• T ∗ is a projection onto the second summand in the theorem (complex Hermitian withnonsimple envelope), then by corollary 4.32:

T∞(A) =

K2∑k=1

2∑o,p=1

λ(p)i V (p)

i

(tr(o)

i,2

[V (o)∗

i AV (o)i

]⊗ ρ

(p)i

)δopTV (p)∗

i

with δopT means that we take the transposition iff p , o.

• T ∗ is a projection onto the third summand in the theorem (symmetric elements), then bycorollary 4.28:

T∞(A) =

K3∑k=1

Vk

trk,2

V†k AVk + (V†k AVk)T

2

⊗ ρk

V†k

• T ∗ is a projection onto the fourth summand in the theorem (quaternionic Hermitian ele-ments), then by corollary 4.30:

T∞(A) =

K4∑k=1

Vk

trk,2

V†k AVk + (Q ⊗ 1mk )−1(V†k AVk)T (Q ⊗ 1mk )

2

⊗ ρk

V†k

• T ∗ is a projection onto the fifth summand in the theorem (embeddings of spin factorsVk

with k = 2n), then by corollary 4.14:

T∞(A) =

K5∑i=1

ki∑j=1

12ni

Vi

(S σ0

j ⊗ ρi tri(Sk j,σ0j V†i AVi)

)V†i

102

Martin Idel The peripheral spectrum of positive, trace-preserving maps

• T ∗ is a projection onto the sixth summand in the theorem (embeddings of spin factorsVk

with k = 2n), then by corollary 4.16:

T∞(A) =

K∑i=1

ki∑j=0

2∑o,p=1

λ(o)i

2niV (o)

i

(S ki,σ0,(o)

j

(12ni ⊗ ρ

(o)i

))V (o)∗

i tr(p)i

[V (p)∗

i AV (p)i S ki,σ0,(p)

j

]Here, all ρ are nonsingular states and all λ ∈ (0, 1), where λ(1)

i +λ(2)i = 1 always (for specifications,

see the cited corollaries).But then the image of these projections is given by algebras exactly like the one above.Finally, we need to get rid of our assumption that T has a full-rank fixed point. All technicalideas to do so were already introduced in the proof of corollary 2.27.Let T :Md →Md be a positive, trace-preserving map. We know that for X ∈ FT , X has supportand range in T∞(1) by lemma 2.7. Call Q the projection onto the support space of T∞(1) andH the support space itself. Then define V : H → Cd an isometry such that VV† = Q. DefiningT : B(H)→ B(H) via T (X) = V†T (VXV†)V , we have that T is a positive map with a full rankfixed point. It is also trace-preserving, since tr(T (X)) = tr(V†T (VXV†)V) = tr(T (VXV†)Q) =

tr(T (VXV†)) = tr(VXV†) = tr(X) using that T is trace-preserving and V is an isometry. HenceFT † is a unital Jordan algebra. However, since (as in the proof of corollary 2.27)

T (X) = 0 ⊕ T (X) ∀X : QXQ = X

in a specific basis (the basis change may be incorporated into the unitary U), we have that inparticular FT = 0 ⊕ FT . But this is exactly the structure as claimed.Since T∞(1) is the fixed point with maximal rank, ρ is as claimed in equation 6.2. The spaceρ−1/2FTρ

−1/2, where the inverse is taken on the support of ρ, now has the structure of theorem3.3, hence is a Jordan algebra, as claimed.

This theorem now completely solves the problem of fixed points of positive, linear maps (trace-preserving or unital) which do not satisfy the Schwarz inequality, which was also posed asproblem 13 in [40].

6.2. The peripheral spectrum of positive, trace-preserving maps

Section 6.5. of [40] is concerned with the peripheral spectrum (or more precisely its linear span)of a positive, trace-preserving map (see definition 2.4). Since the theorems there are more or lesscorollaries to the theorem on fixed point spaces, we will give them here, too.

Proposition 6.3 ([40], prop. 6.12). Let T :Md →Md be a positive, trace-preserving map. Then

1. Tφ(Md) = XT

2. ∃ρi ≥ 0 : XT = spanρi

103


3. T (XT ) = XT

Proof. 3. holds, since any X ∈ XT is a linear combination of peripheral eigenvectors of T , butthen X will be mapped to a linear combination of peripheral eigenvectors of T , hence T (X) ∈ XT .1. follows by using lemma 2.6 and realizing that XT is then indeed the complete fixed point spaceof T . 2. follows from 1. by realizing that Tφ (by virtue of equation (2.6)) is still trace-preservingand positive and the fixed point space of a positive, trace-preserving map is spanned by positiveelements.

With this in mind, we now know that XT has the structure as in theorem 6.1:

Theorem 6.4. Let T :Md →Md be a positive, trace-preserving map. Let XT be the linear spanof the peripheral spectrum of T . Then XT has the structure of theorem 6.1, in particular, it is asemi-simple Jordan algebra with respect to a modified product.We denote the different types of Jordan algebras as in theorem 3.3. If we assume that T is unital(in the sense of the modified Jordan product, e.g. T (0⊕

⊕1Ji ⊗ ρi) = 0⊕

⊕1Ji ⊗ ρi, if we only

have Jordan algebras with simple envelope), then we obtain:

• Let XT = 0 ⊕⊕K

i=1Ji ⊗ ρi with Ji a Jordan algebra with simple envelope, then everyX ∈ XT has the form

X = 0 ⊕K⊕

i=1

xi ⊗ ρi

with xi ∈ Ji. Then we have:

T (X) = 0 ⊕K⊕

i=1

Ti(xπ(i)) ⊗ ρi

with some permutation π which permutes within the subsets of 1, . . . ,K for which Jk

have equal dimensions and Ti : Ji → Ji a *-isomorphism or *-antiisomorphism (i.e.Ti(x · y) = Ti(x) · Ti(y) or Ti(x · y) = Ti(y) · Ti(x) and Ti is a linear isomorphism).

• In the same setting as the last item, if Ji =Md, then Ti(x) = Uix(T )U†i with unitaries Ui

and (T ) indicating that a transposition might occur.

• Let XT = 0⊕⊕K

i=1Ji with Ji =Mdi(C)⊗ ρi or Ji = diag(A⊗ ρi(1) , AT ⊗ ρi(2)) | A ∈ Mdi.Then every X ∈ XT has the form

X = 0 ⊕K1⊕i=1

xi ⊗ ρi ⊕

K⊕i=K1+1

(λixi ⊗ ρi(1) ⊕ (1 − λi)xT

i ⊗ ρi(2)

)

104


with xi ∈ Mdi and some K1 ≤ K. Then we have:

T (X) = 0 ⊕K1⊕i=1

Uix(T )π(i)U

†

i ⊗ ρi ⊕

K⊕i=K1+1

(λiUix

(T )π(i)U

†

i ⊗ ρi(1) ⊕ (1 − λi)Ui(xTπ(i))

(T )U†i ⊗ ρi(2)

)where again π is some permutations permuting within the subsets of 1, . . . ,K wherethe summands have equal dimension. Furthermore, Ui are again some unitaries and (T )

indicates that a transposition might occur.

• Let XT = 0 ⊕⊕K

i=1Ji with Ji = Jspinkifor ki odd and either simple or nonsimple

envelope. Then the maps look similar to the last scenario. In particular, let S 1, . . . , S ki

denote the embedded spin system of the nonuniversal embedding of proposition 3.8, thenthe universal embedding is obtained by S 1 := S 1 ⊗ 1, . . . , S k−1 := S k ⊗ 1, S k = S k ⊗ σ3.Then XT is spanned by

X = 0 ⊕K1⊕i=1

S i ⊗ ρi ⊕

K⊕i=K1+1

(λiS i ⊗ ρi(1) ⊕ ±(1 − λi)S T

i ⊗ ρi(2)

)with some K1 ≤ K and the ± referring to the sign of S i. Then we have:

T (X) = 0 ⊕K1⊕i=1

Ti(S π(i)) ⊗ ρi ⊕

K⊕i=K1+1

(λiTi(S π(i)) ⊗ ρi(1) ⊕ ±(1 − λi)Ti(S π(i)) ⊗ ρi(2)

)where again π is some permutations permuting within the subsets of 1, . . . ,K where thesummands have equal dimension and Ti is either a *-isomorphism or a *-antiisomorphismof the nonuniversal embedding ofVk and the ± refers to the sign of S .

• All items can be combined without obtaining new cases. For readability, we refrain fromdoing so.

Proof. The first part follows directly from proposition 6.3. For the rest, we need a more involvedstrategy. In Step 1, we prove that T has an inverse on XT . Then, in Step 2, we will see thatbecause of this, T must map minimal projections to minimal projections. This in turn, togetherwith continuity and dimensional arguments implies (Step 3) that the map must send any blockin the direct sum to a block of the same size and type of Jordan algebra. This (Step 4) againimplies that T must restrict to a Jordan isomorphism on the different summands (modulo somepermutation).Step 1: We first take some ideas from [40], to prove that T has an inverse on XT . To this end, letlimi→∞

T ni = Tφ for some ni increasing subsequence. Then Tφ is also positive and trace-preserving

and furthermore, since XT is the fixed point space of Tφ, Tφ acts as identity on XT (see prop. 6.3again). Now T−1 := lim

i→∞T ni−1 is the inverse of T on XT , since T−1T = Tφ = 1 when restricted

105


to XT . Since T−1 is the limit of a positive, trace-preserving map, it is itself again positive andtrace-preserving.From now on, the proof differs from [40]. Consider first the case where all envelopes are simple.Step 2: T must map minimal projections to minimal projections:By construction, X ∈ XT , then

X = 0 ⊕K⊕

i=1

xi ⊗ ρi

We know from lemma A.3 that the Jordan algebra is spanned by its minimal projections. Leti ∈ 1, . . . ,K be arbitrary and consider a minimal projection pi ∈ Ji, then Pi = 0⊕pi⊗ρi ∈ XT⊕0is a minimal projection. We know:

T (Pi) = 0 ⊕K⊕

j=1

x j ⊗ ρ j =: Xi

for some xi ∈ Ji. In particular, since T is positive and trace-preserving, tr(Xi) = tr(Pi) and Xi ≥ 0,hence we have a spectral decomposition Xi =

∑j λ j (0 ⊕ P j ⊗ ρi j ⊕ 0) with i j ∈ 1, . . . ,K always

and P j ∈ Ji j a minimal projection. Observe

Pi = T−1(Xi) =∑

j

λ jT−1(0 ⊕ P j ⊗ ρi j ⊕ 0)

since T is bijective on XT . The positivity implies in particular

T−1(0 ⊕ P j ⊗ ρi j ⊕ 0) ≤ Pi ∀ j

However, Pi was chosen minimal, hence T−1(0 ⊕ P j ⊗ ρi j ⊕ 0) ∝ Pi and since the map is trace-preserving, we have equality. Now T is bijective, hence 0 ⊕ P j ⊗ ρi j ⊕ 0 = 0 ⊕ Pk ⊗ ρik ⊕ 0for all j, k, since their images coincide. But then Xi =

(∑j λ j

)0 ⊕ P ⊗ ρ j0 ⊕ 0 for some j0

and some minimal projection P ∈ J j0 . If the two projections P and Pi have the same rank,then, since T is trace-preserving,

∑j λ j = 1 and a minimal projection is mapped to a minimal

projection, otherwise a minimal projection is mapped to some Xi which is proportional to aminimal projection with prefactor λi = tr(Pi)/ tr(Xi) ∈ R+, hence we have nearly proved thatminimal projections map to minimal projections. The final step will be done at the end of step 3.Step 3: We claim that for X = 0 ⊕

⊕Ki=1 xi ⊗ ρi with xi ∈ Ji we have:

T (X) = 0 ⊕K⊕

i=1

Ti(xπ(i)) ⊗ ρi

where π is a permutation that permutes thoes blocks of the same type and size of Jordan algebra.We already saw that any minimal projection is mapped into one summand of the peripheral

106


spectrum, hence elements of the form 0⊕ xi⊗ρi⊕0 for some block i ∈ 1, . . . ,Kmust be mappedto the same block i′ (i′ may be different from i but all elements must be mapped to the same blocki′) for all xi ∈ Ji by continuity of T . Since T is linear and invertible, this already proves that onlyblocks of the same dimension can be mapped to each other, hence we obtain a permutation πbetween the blocks of same dimension as in the theorem. This means we already know that

T (X) = 0 ⊕K⊕

i=1

Ti(xπ(i)) ⊗ ρi

where π might also commute between Jordan algebras of differnt type but same dimensionas algebra (e.g. the spin factor V63 and M8 have the same dimension) and the Ti must bepositive, linear maps. To see that π can only permute within the set of automorphic algebras, letTi : Jπ(i) → Ji be the map in the decomposition of T , then Ti is invertible and linear, since it isjust the restriction of an invertible, linear map to some subspace. Since Jπ(i) and Ji are unitalJordan algebras, we can decompose their units into a set of minimal projections:

1Jπ(i) =

n∑j=1

P j 1Ji =

n∑j=1

P j

and by lemma A.10, we know that n and n are well-defined. By considering either Ti or T−1i , we

can assume that n ≤ n. Now Ti(1) =:∑n

j=1 Ti(Pi) ∈ Ji. By positivity of Ti, we can then find aspectral decomposition of Ti(1) =

∑mj=1 λiQi where m ≤ n, since the Qi are mutually orthogonal

projections. But then, since minimal projections are sent to minimal projections (modulo someprefactor), 1 = T−1

i (Ti(1)) =∑m

j=1 λiλQi, by definining λQi := T−1i (Qi). Then Qi , Q j for i , j

by the bijectivity of T and the Qi are minimal projections. However, m ≤ n ≤ n and if n < n,then this cannot be, since we have a decomposition of the unit in Ji into minimal projectionswith less projections than possible. Hence m = n = n. This implies that the Jordan algebras mustbe of the same type (see corollary A.11). This in turn gives that π may only commute betweenautomorphic Jordan algebras, because it can only commute between Jordan algebras of the sametype that have the same dimension. Given a dimension and type, however, there is at most oneJordan algebra of this type by lemma A.12. Since we have seen that the Jordan algebras mustbe of the same type, this implies that π can only permute between isomorphic algebras, hencewe have the structure as in the claim, where Ti are isomorphisms. This also proves that minimalprojections are mapped to minimal projections, since minimal projections of isomorphic algebraswith envelopeMd for some d ∈ N have the same rank via A.10.Step 4: Finally, we have to prove that the Ti from the last step are actually Jordan isomorphisms.To this end recall that we have seen that minimal projections are mapped to minimal projections.We assumed that T (0 ⊕

⊕Ki=1 1 ⊗ ρi) = 0 ⊕

⊕Ki=1 1 ⊗ ρi, hence the Ti are unital. Now consider

mutually orthogonal minimal projections in the same Jordan block, P1, P2 ∈ XT . Then Ti(P1 +

P2) = P1 + P2 where P1, P2 are minimal projections. We have:

(P1 + P2)2 = P1 + P2 + 2P1 ∗ P2

107


Since (P1 + P2)2 = P1 + P2 since P1, P2 are orthogonal, by the Jordan-Schwarz inequality (lemma2.24), we obtain

P1 + P2 = Ti((P1 + P2)2) ≥ Ti(P1 + P2) ∗ Ti(P1 + P2) = P1 + P2 + 2P1 ∗ P2

⇔ 0 ≥ P1 ∗ P2

however, this is not possible, since Pi are projections. We need to see that P1 ∗ P2 must haveat least one positive eigenvalue (if P1 ∗ P2 , 0). Observe that all entries on the diagonal of P2must be positive, since otherwise P2 cannot be an idempotent. In addition, if we choose the basissuch that P1 is diagonal, we can see that the trace of P1 ∗ P2 must be nonnegative, hence if eitherP1 ∗ P2 has a positve eigenvalue or P1 ∗ P2 = 0. Since the first case of positive eigenvalues isexcluded, the two projections must be orthogonal.An iterative argument thus shows that Ti maps projections to projections. However (see e.g.[24]), this implies that Ti is a Jordan isomorphism. To see this, note that if the equality in theJordan-Schwarz-inequality holds for all projections (as we have just seen), then using the spectraldecomposition, it holds for all positive elements. Since XT is the fixed point space of the positive,trace-preserving map Tφ, it is spanned by its positive elements via lemma 2.7, hence decomposingan arbitrary A ∈ Ji into four positive elements, Ti(A ∗ B) = Ti(A) ∗ Ti(B) for all B ∈ Ji and all iby using lemma 2.25. This proves the assertion for Jordan algebras with simple envelopes.Special case: The special case of Ji =Mdi is obtained as in [40], using that any positive, trace-preserving map with a positive inverse must be of the form Tk(x) = UxU† or Tk(x) = UxT U† forsome unitary U (this can be seen e.g. from Wigner’s theorem).Now we consider the case where Ji = JC orMd. The proof is analogous to the proof above,noting that the maps T : Md → J

Cd ; A 7→ diag(UAU†,U(A)T U†)(T ) are positive and can send

C*-algebras to embeddings of the type JC bijectively. Similarly, one obtains the proof for thecase of spin factors.

The proof as it is presented here is slightly weaker than the theorem, since the form of theJordan-isomorphism is a bit more general than the form forMd. To obtain the full result, weneed another lemma:

Lemma 6.5 ([13], Theorem I). Any Jordan-*-automorphism between two simple algebras iseither a *-automorphism or a *-antiautomorphism.

Proof. The proof is very technical and lengthy, see [13] or [24] for further reference.

This lemma then completes the proof of theorem 6.4. Apart from our usual assumptions (T ispositive and trace-preserving), we have one additional assumption in this theorem, namely that Tmaps 0⊕

⊕Ki=1 1⊗ρi to itself. For completely positive maps, this assumption is not needed, since

the results are obtained on a different path (see [40], section 6.5), but the maps turn out to obeythis unitality anyway. Likewise, we expect that the assumption is superfluous here. However,in our proof we need the unitality to show that mutually orthogonal projections are mapped

108


to mutually orthogonal projections. Thus far, we have not found a way to either circumventthe use of the Jordan-Schwarz-inequality in this step or prove the unitality of T differently andbeforehand.This finishes the discussion about extensions of the results presented in [40].

109

Conclusion

Finally, let us review our results. In chapter 1, we extended a normal form, which relates certainnonnegative matrices to doubly stochastic matrices to a normal form, which relates certainpositive maps to doubly stochastic maps. Roughly speaking, our main theorem 1.15 givessufficient conditions for a positive map to be equivalent to a direct sum of doubly stochasticmaps in the sense that if T is the positive map, then XT (Y(·)Y†)X† is roughly a direct sum ofdoubly stochastic maps. We also conjecture necessary and sufficient conditions in conjecture1.26, thereby introducing the notion of a ”fully indecomposable“ map.The main part of this thesis was about fixed points of positive, trace-preserving maps. Werecall that the previous results were concerned with completely positive maps. In fact, given acompletely positive, trace-preserving map T :Md →Md, it was shown (e.g. [40]) that the fixedpoint algebra is isomorphic to

FT = 0 ⊕n⊕

i=0

Mdi ⊗ ρi

with ρi some states. This was obtained from considering unital maps T ∗ were T has a full-rankfixed point. In this case, the fixed point algebra turned out to be a C*-algebra and the fixed pointprojection T ∗∞ must therefore be a projection onto a C* algebra, characterized by proposition4.25.In this thesis, we now extended the result to arbitrary linear, positive maps that are either unitalor trace-preserving. We first saw that the fixed point spaces of unital, positive maps are more orless (special) Jordan algebras (theorem 2.26 + corollary). Then we characterized all embeddingsof Jordan algebras in theorem 3.3, which follows as a straightforward calculation from someolder results. We also constructed all positive, unital projections onto Jordan algebras in theorem4.1, which allows us to classify fixed point spaces of positive, trace-preserving maps. Most ofthe positive, unital projections were already well known in the literature, while it seems that notall projections onto spin factors were classified, yet. In this sense, added some results for finitedimensional Jordan algebras, completing the theory. In summary, we obtained the analogousresult for fixed point spaces of positive, trace-preserving maps, proving that they are given by:

FT = U

0 ⊕ K1⊕

i=1

Mdi ⊗ ρi

⊕ K2⊕

i=1

JCdi

⊕ K3⊕

i=1

JTd′i⊗ ρ′i

⊕

K4⊕i=1

JHd′′i⊗ ρ′′i

⊕ K5⊕

i=1

Jspinfullki⊗ ρ′′′i

⊕ K6⊕

i=1

Jspinsumki

U†

110


where U ∈ Md is a unitary and 0 < ρi, ρ′i , ρ′′i , ρ

′′′i are states. The different termsJC,T,H andJspin

correspond to different kinds of embeddings of different types of Jordan algebras, characterizedin theorem 2.13. As an application of the knowledge about fixed point algebras, we proved atheorem about positive, trace-preserving maps on the peripheral spectrum (theorem 6.4).In addition, we also saw that the projections onto these algebras are not always decomposable, infact, they are only decomposable, if the algebra is not an embedding of a spin factor of sufficientdimension (see theorem 5.1. These results date back some time (see [37] for a review), however,we were not aware of them at the beginning of the thesis.Finally, let us give a small outlook. One could ask what happens with arbitrary positive maps.We have seen in lemma 2.7 that the fixed point space of arbitrary maps is at least a vectorspace. However, for an arbitrary positive map T , the case is more difficult. As long as T ∗ hasa full-rank positive fixed point (the space of fixed points is no longer necessarily spanned bypositive operators), the fixed point space will again be a Jordan algebra, however not unital and,maybe, not necessarily semi-simple. However, a classification of possible fixed point spaces forarbitrary positive maps might also be too general to be of use.Another question would be to consider infinite dimensional Jordan algebras. Here, the answeris simpler, because most of the results (especially the decomposability result in chapter 5) wereoriginally formulated in infinite dimensions. It turns out that the fixed point space of positive,continuous, unital and linear map turns out to be a so-called JW-algebra or a JC-algebra, dependingon the continuity properties of T . JW-algebras however look a lot like finite-dimensional Jordanalgebras and the abstract results for reversible algebras carry over immediately since they, too,were formulated in infinite dimensions (see [34, 35, 8, 36, 12]). The only irreversible algebra ofinfinite dimensions is then the CAR-algebra, which is heavily studied in the literature, due to itsimportance in fermionic physics.

111

Bibliography

[1] R. Bhatia. Positive Definite Matrices. Princeton University Press, 2007.

[2] O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics 1.Springer, 1987.

[3] H. Braun and M. Koecher. Jordan-Algebren. Springer, 1966.

[4] R. A. Brualdi, S. V. Parter, and H. Schneider. The diagonal equivalence of a nonnegativematrix to a stochastic matrix. Journal of Mathematical Analysis and Applications, 16:31–50,1966.

[5] H. Chen. Projections and types of von neumann algebra. http://www.eleves.ens.fr/home/gheysens/gnc/gnc_8_huan.pdf, 2012. Seminar.

[6] C.-H. Chu. Jordan Structures in Geometry and Analysis. Camebridge University Press,2012.

[7] R. G. Douglas. On majorization, factorization, and range inclusion of operators on hilbertspace. Proc. Amer. Math. Soc., 17(2):413–415, 1966.

[8] E. Effros and E. Størmer. Positive projections and jordan structure in operator algebras.Math. Scand., 45:127–138, 1979.

[9] L. Gurvitz. Classical complexity and quantum entanglement. Journal of Computer andSystem Sciences, 69(3):448–484, 2004.

[10] K.-C. Ha. Positive projections onto spin factors. Linear Algebra and its Applications,348:105–113, 2002.

[11] W. Hall. Constructions of indecomposable positive maps based on a new criterion forindecomposability. Journal of Physics A, 2006.

[12] H. Hanche-Olsen and E. Størmer. Jordan Operator Algebras. Pitman Advanced PublishingProgram, 1984.

[13] I. N. Herstein. Jordan homomorphisms. Transactions of the AMS, pages 331–341, 1956.

[14] M. Horodecki, P. Horodecki, and R. Horodecki. Separability of mixed states: Necessaryand sufficient conditions. Physics Letters A, 223(1-2):1–8, 1996.

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[15] F. D. Jacobson and N. Jacobson. Classification and representation of semi-simple jordanalgebras. Transactions of the AMS, 65(2):141–169, 1949.

[16] N. Jacobson. Structure and Representations of Jordan Algebras. AMS, 1968.

[17] P. Jordan, J. von Neumann, and E. Wigner. On an algebraic generalization of the quantummechanical formalism. Annals of Mathematics, 35(1):29–62, 1934.

[18] S. Lang. Algebra. Springer, corrected version 2005.

[19] B. Lemmens and R. Nussbaum. Nonlinear Perron-Frobenius Theory. Cambridge UniversityPress, 2012.

[20] G. Letac. A unified treatment of some theorems on positive matrices. Proc. Amer. Math.Soc., 43(1):11–17, 1974.

[21] D. London. On matrices with a doubly stochastic pattern. Journal of Math. Analysis andApplications, 34:648–652, 1971.

[22] K. McCrimmon. A Taste of Jordan Algebras. Springer, 2004.

[23] M. V. Menon. Reduction of a matrix with positive elements to a doubly stochastic matrix.Proc. Amer. Math. Soc., 18(2):244–247, 1967.

[24] L. Molnar. Selected Preserver Problems on Algebraic Structures of Linear Operators andon Function Spaces. Springer, 2007.

[25] P. Muhly and J. N. Renault. C*-algebras of multivariate wiener-hopf operators. Trans. Amer.Math. Soc., 274(1), 1982.

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A. Appendix

A.1. Spectral and Minimal Projections in Jordan Algebras

This section is devoted to the study of projections within (special, unital) Jordan algebras. Sincethis is not central to the understanding of the main part of the thesis, we provide it here as anappendix. We do not claim to have discovered anything new in this section. In principle, theresults below follow from the structure theory of so-called JBW-algebras and especially thestructure theory for its lattice of projections and the fact that it is modular in finite dimensions(see chapter 5 in [12] or [6]). However, as becomes clear, this approach requires the introductionof many different objects and even more basic theory, hence we will develop the results weneed along more elementary lines. The proofs are not taken specifically from any source but areinspired by [5, 12] and [22].The structure is as follows: After recalling the basic definitions of minimal and orthogonalHermitian projections, we essentially want to prove that any Jordan algebra contains all itsspectral projections (lemma A.3) and for any two orthogonal projections P1, P2 ∈ J , where J isa simple algebra, there exists another Hermitian projection P3 ≤ P1 + P2 different from P1, P2(proposition A.9). In passing, we note how these projections look like (lemma A.7). Finally, wediscuss minimal projections in spin factors.

Definition A.1. Let J be a Jordan algebra.

• P ∈ J is a projection if P2 = P.

• P ∈ J is called an Hermitian projection, if it is a projection and P† = P. From now on,every projection will be assumed to be Hermitian.

• Two projections P1, P2 ∈ J are called orthogonal, if P1 ∗ P2=0

• 0 , P ∈ J is called minimal if there does not exist a nonzero projection P ∈ P such thatP ≤ J .

Any self-adjoint A ∈ J , A has a spectral decomposition

A =

n∑i=1

λiPi (A.1)

with λi ∈ R and∑n

i=1 Pi = 1, Pi ∈ S(J) and Pi ∗ P j = 0 for all i , j.

115

Martin Idel Spectral and Minimal Projections in Jordan Algebras

In a special Jordan algebra, this implies that any projection P ∈ J is also a projection of theenvelope. Likewise, two orthogonal projections in the algebra are orthogonal in the envelope, i.e.P1P2 = P2P1 = 0. Let us also introduce

Definition A.2. Let X,Y,Z ∈ J a Jordan algebra. Then define:

UX(Y) := 2X ∗ (X ∗ Y) − X2 ∗ Y =12

X ∗ (XY + YX) − X2 ∗ Y

=14

(X2Y + XYX + XXYX + YX2) − X2 ∗ Y

= XYX

X,Y,Z := 2(X ∗ (Y ∗ Z) + (X ∗ Y) ∗ Z − (X ∗ Z) ∗ Y)

=12

(XYZ + XZY + YZX + ZYX + XYZ + YXZ + ZXY + ZYX − (X ∗ Z) ∗ Y)

= XYZ + ZYX

Note that UX(Y) = 12 X,Y, X.

First, we need a lemma, whose formulation we take from [25], which refers to [3] for a proof.We will not take this route, but instead use tools from the theory of von Neumann algebras aspresented in chapter 2.4.2. in [2]:

Lemma A.3. Let J be a Jordan algebra and A ∈ J a self-adjoint element. Then all its spectralprojections also lie in J .

Proof. We have already noted that the definition of a Jordan algebra implies that it is power-associative, i.e. the Jordan subalgebra A generated by 1 and A is associative. As it is a finitedimensional algebra, it is necessarily a closed C*-algebra, hence in particular a von Neumannalgebra. However, any von Neumann algebra contains all its spectral projections, as is proved in[2].

In finite dimensions, it is obvious that minimal projections always exist. We have even more:

Lemma A.4. Let J be a Jordan algebra (C*-algebra) and P1 ∈ J an orthogonal projections,then there exist minimal projections Pi such that

∑i Pi = P and Pi ∗ P j = 0.

Proof. The proof proceeds iteratively. Let P1 ≤ P be a minimal projection with P1 ∈ J . Then0 ≤ P − P1 ∈ J is again a projection, since P1 ∗ P = P1, since P1P = PP1 = P1 for P1 ≤ P andtherefore (P−P1)2 = P∗P−P1∗P−P∗P1 +P1∗P1 = (P−P1). Also, P1∗(P−P1) = P1−P1 = 0hence P1 and P − P1 are orthogonal. Now if P − P1 is already minimal, then we are done, elsewe repeat the procedure.

Another question one can ask is whether, given two minimal and orthogonal projections P1, P2 ∈

J , there is a third minimal projection P3 ≤ P1 + P2 but P3 Pi for i ∈ 1, 2, i.e. whether we

116


can decompose P1 + P2 into a sum of different minimal orthogonal projections. The questionmakes only sense, when there are two nonzero orthogonal minimal projections in the Jordanalgebra. However, in finite dimensions, when this is not the case, the algebra must be trivial andwe implicitly exclude this case from further discussion.To answer the question, we first need a decomposition of the Jordan algebra:

Proposition A.5 (Peirce decomposition for special Jordan algebras, [22]). Let J be a Jordanalgebra and consider projections Pi for i = 1, . . . , n with a nonnegative n and

∑ni=1 Pi = 1 and

Pi ∈ J are pairwise orthogonal. Then we have a decomposition

J =⊕i≤ j

Ji j (A.2)

where Jii := PiJPi and Ji j := PiAP j + P jAPi | A ∈ J and the Jii are Jordan subalgebras ofJ .

This is a special case of the so-called Peirce decomposition for general Jordan algebras (see [16]or [22]). The proof also works for C*-algebras, where we could further decompose J12.

Proof. Recall that we have for X,Y,Z ∈ J

UX(Y) =12

(X2Y + XYX + XYX + YX2 − X2Y − YX2) = XYX

X,Y,Z = XYZ + ZYX

In particular, X,Y,Z,UX(Y) ∈ J . Now consider Pi ∈ J for i = 1, . . . , n as described above,then for all A ∈ J

A =

n∑i=1

Pi

A

n∑j=1

P j

=

n∑i=1

PiAPi +∑i< j

PiAP j +∑j<i

PiAP j

=

n∑i=1

PiAPi +∑i< j

(PiAP j + P jAPi)

=⊕i≤ j

Ji j

In particular, PiAPi = UPi(A) ⊂ J and PiAP j + P jAPi = Pi, A, P j ∈ J for all A ∈ J . Hencefollows the claim. Note that we do not claim that Ji j , 0. Finally, Jii are subalgebras, since

PiAPi ∗ PiBPi = Pi(A ∗ PiBPi)Pi ∀A, B ∈ J

which implies the claim.

117


Corollary A.6. Let J be a Jordan algebra. Given P1, P2 orthogonal projections, we can find aprojection 0 , P3 ≤ P1 + P2 and P3 different from P1, P2 if J12 , 0 in the corresponding Peircedecomposition (A.2).

Proof. If J12 , 0, then we can pick 0 , A ∈ J12 self-adjoint (recall J12 = (J12)h + i(J12)h).Consider the spectral decomposition A =

∑i λiPi +0 · (1− (P1 + P2)) (since A ∈ J12) with λi , λ j

for i , j. By assumption,∑

i Pi = P1 + P2 and by lemma A.3, P1 ∈ J . Obviously P1 , P1 andP1 , P2 since otherwise, A < J12, since P1 ∈ J11 and P2 ∈ J22.

Now, let us have a look, how these projections would look like.

Lemma A.7. Let J be a Jordan algebra (C*-algebra). Let P1, P2 be minimal orthogonalprojections in J . Let P3 ≤ P1 + P2 be a minimal projection in J different from P1 and P2.Restricting to P1JP1, in the basis where P1 = diag(1, 0) and P2 = diag(0,1), we have

P3 =

(α1 βBβB† γ1

)(A.3)

with α + γ = 1, α, γ ≥ 0, |β|2 = αγ and B†B = 1, BB† = 1. In particular, this implies that therank of P1 and P2 must be equal.We can also write

P3 = αP1 + γP2 + βV + βV† (A.4)

with partial isometries V such that VV† = P2 and V†V = P1. In particular, βV + βV† ∈ J .

Proof. Let P3 =

(A BC D

). Since P3 must be Hermitian, C = B† and A = A†, D = D†. In particular,

using the Peirce decomposition (A.2), A,D ∈ J . Now A ∝ 1, since otherwise A =∑

i λiQi

a spectral decomposition with λi , 0 for some i and λi , λ j for some i , j. By lemma A.3,spectral projections of A ∈ J belong to J , but then, since necessarily Qi ≤ P1, this contradictsthe minimality of P1, hence A ∝ 1. Likewise, D ∝ 1.This means we can write:

P3 =

(α1 BB† γ1

)and α, γ ∈ R since P†3 = P3. Now P3 is supposed to be a projection, in particular:

P23 =

(α21 + BB† αB + γB

αB† + γB† γ21 + B†B

)=

(α1 BB† γ1

)= P3

By assumption, B , 0, since otherwise P3 is a linear combination of P1 and P2, hence notminimal. Then B†B , 0, thus B†B ∝ 1, since otherwise γ2

1 + B†B , γ1. Likewise, BB† ∝ 1,

118


which is only possible if the two identities have the same rank, i.e. P1 and P2 have the same rank.Furthermore, α+ γ = 1, since (α+ γ)B = B, which implies that if we set B := B/β with |β|2 = αγ,then P2

3 = P3, via α2 + |β|2 = (α + γ)α = α.

This proves the claim. In particular, V :=(0 B0 0

)is a partial isometry fulfilling VV† = P1 and

V†V = P2 and βV + βV† ∈ J by the Peirce decomposition (A.2).

Finally, we can state the following result that tells us that such projections exist in a simple Jordanalgebra:

Corollary A.8. Given a simple Jordan algebra J and a decomposition of its unity into minimalorthogonal projections 1 =

∑ni=1 Pi with n ∈ N, then there exists a j ∈ 2, . . . , n and a minimal

projection P with P ≤ P1 + P j and different from P1 and P j.

Proof. We use the Peirce decomposition (A.2): J =⊕

i≤ jJi j. Since J is simple, there exists aj ∈ 2, . . . , n such that J1 j , 0, otherwise J = P1JP1 ⊕ P⊥1JP⊥1 , where P⊥1 :=

∑ni=2 Pi fulfills

P1 + P⊥1 = 1. Corollary A.6 now guarantees the existence of P.

This is the last preparation we need for:

Proposition A.9. Given a simple Jordan algebra J and two minimal projections P1, P2 ∈ J ,then there exists a third minimal projection P ∈ J with P ≤ P1 + P2 and P1 , P , P2.

Proof. We start from a decomposition of unity into mutually orthogonal, minimal projections,i.e. 1 =

∑ni=1 Pi, where

∑ni=3 Pi = 1 − P1 − P2 defines the decomposition given P1, P2. Using

corollary A.8, we know that there exists P j, j , 1 such that there exists a minimal projection P j ∈

J11⊕J1 j⊕J j j different from P1 and P j. We know from lemma A.7 that P j = αP1+γP j+βV+βV†

with βV + βV† ∈ J a partial isometry with VV† = P1 and V†V = P j.If j = 2, we are done, hence suppose j , 2.Now suppose there exists a projection P2 ∈ J22 ⊕ J2 j ⊕ J22, which is nonzero and differentfrom P2 and P j. Then again P2 = αP2 + γP j + βW + βW† and βW + βW† ∈ J2 j. In particular,we have WW† = P2 and W†W = P j. Then, looking at the block-diagonal matrix structure, wehave 0 , (V + V†) ∗ (W + W†)† = WV† + VW† ∈ J12 and

WV†(WV†)† = WV†VW† = WP jW† = WW†WW† = P22 = P2

(WV†)†WV† = VW†WV† = VP jV† = VV†VV† = P21 = P1

using the structure of V and W in lemma A.7. Since WV† + VW† , 0, J12 , 0 and hence bycorollary A.6, we can then construct a projection P ∈ J with P ≤ P1 + P2 and P different fromP1 and P2.If there is no projection P2 as above, we need to iterate, finding a k , 1, 2, j such that there existsa projection Pk ≤ P j + Pk which is not equal to P j and Pk. Such a k exists, because otherwiseJ = (P1 + P2 + P j)J(P1 + P2 + P j) ⊕ (P1 + P2 + P j)⊥J(P1 + P2 + P j)⊥, which is impossible

119


as J is simple. Via corollary A.6 and the argument above, we can then also find a projectionPk ≤ P1 + Pk which is not equal to P1 and Pk in the same manner. Iterating and using thesimplicity of J then gives the result. Since n is finite, the iteration terminates.

We need another lemma:

Lemma A.10. Let J be a Jordan algebra and 1 =∑n

i=1 Pi a decomposition into mutuallyorthogonal minimal projections. Then given another decomposition 1 =

∑ni=1 Pi into mutually

orthogonal minimal projections, n = n.

Proof. This result is given in [12], following from lemmata 5.2.1-5.2.3. However, we give a verydifferent proof, which is inspired by some ideas there.First, we note that in the proof of lemma A.7 (together with proposition A.9), we found that twominimal projections that are orthogonal have necessarily the same rank. In particular, all Pi in theabove decomposition have the same rank.Let P be another minimal projection. It suffices to show that P has the same rank as any of thePi, because then, given a decomposition 1 = P +

∑n−1i=1 Pi into mutually orthogonal, minimal

projections, this implies that all Pi have the same rank and hence, as the ranks must add up to thedimension of the algebra, n = n. Now if P is orthogonal to one of the Pi, this follows again by theproof of lemma A.7 together with A.9, since then we can find a Pi in J such that Pi ≤ Pi + P anddifferent from both Pi and P. However, then Pi must look like equation (A.3) and in particular, Pand Pi have the same rank. Hence we can assume that P is nonorthogonal to P1 (and any other Pi

for that matter).Now P1PP1 = P1PP1 , 0 is a self-adjoint element of J . Since P1PP1 ∈ J11 in the Peircedecomposition, this implies P1PP1 = λP1 for some λ , 0, since if P1PP1 =

∑ni=1 λiPi is the

spectral decomposition with λi , 0, then Pi ≤ P1, hence by minimality of P1, Pi = P1. Likewise,PP1P , 0 and PP1P = λP for some λ , 0. However, since matrix multiplication is ranknonincreasing, this already implies rank(P1) = rank(P), which is all we needed to show.

This immediately gives us the following result:

Corollary A.11. Let J be a Jordan algebra with simple envelope such that two orthogonal,minimal projections P1, P2 ∈ J fulfill P1 + P2 = 1. Then J is a spin factor.

Proof. We go through the different cases. It suffices, to consider one embedding of each type,since via equivalence, we obtain the result for all embeddings.Let J =Md(C), then obviously 1 =

∑di=1 Eii, where Eii ∈ J is a matrix unit with 1 at position

(i, i) and 0 elsewhere. Obviously, E2ii = Eii and Eii ∗ E j j = 0 for i , j, hence we have a

decomposition into d mutually orthogonal projections. They are minimal, since they are rank one.This means that onlyM2(C) fulfills the assumptions. However,M2(C) can also be seen as thenonuniversal embedding ofV3, hence it is a spin factor.Let J =Md(R), then likewise,

∑di=1 Eii is a decomposition into mutually orthogonal, minimal

projections and again, only d = 2 fulfills the requirements of the lemma. But M2(R) in its

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Martin Idel Omitted proofs from chapter 1

inclusion induced embedding is equivalent to the universal embedding ofV2.Finally, let J =Md(H), then we can see that in its standard embedding σ0 : Md(H)→M2d(C)given by proposition 3.6, we have a decomposition

1 =

d∑i=1

(Eii + Ed+i,d+i) (A.5)

where Eii + Ed+i,d+i ∈ Jσ0 . Clearly, they are mutually orthogonal and in addition, they are

minimal, since otherwise, they could be further decomposed. Suppose we could decomposethem further, then the resulting projections would be rank 1. We know how these projectionswould look like from lemma A.7, but such projections are not in J , as can be seen from thecharacterization in proposition 3.6. Hence, again we have thatM2(H) is the only possible caseand we have already seen that this case corresponds to an embedding of the spin factorV5.

The proof can easily be adapted to hold for all simple Jordan algebras, since the only remainingcase is the universal embedding ofMd(C), where the unit has a decomposition of two orthogonal,minimal projections iff d = 2, which corresponds to the universal embedding ofV3.This last proof is adapted to our needs, although it does not really capture what happens. In fact,one calls a Jordan algebra of type In, if it has a decomposition into n mutually orthogonal minimalprojections. Then one can show that the Jordan algebras of type I2 are precisely the spin factors.We have shown one direction, which is all we need; a more general proof can be found in [35].

Lemma A.12. For every (n,m) ∈ N2, there is at most one Jordan algebra with simple envelope,such that its unit can be decomposed into m minimal projections and that is of dimension n.

Proof. We need the extended theorem from [35], telling us that all spin factors are of type I2.Then we already know that given a dimension, there is only one spin factor with this dimensionand hence the lemma holds for (n, 2). For m ≥ 3, we have the different types of Hermitianmatrices. Here, we have a look at the dimensions. Naturally, the real dimension ofMd is 2d2.The real dimension of JT

d is d(d + 1), and 2m2 , n(n + 1) for all n,m ∈ N and n ≥ 2. Also, aquick calculation shows that the real dimension of JH

d ⊂ M2d is 2d(2d− 1) and 2d(2d− 1) , 2n2

for all n ≥ 2. However, n(n + 1) = 2d(2d − 1) if we set 2 = 2d − 1, hence JT2d−1 and JH

d haveequal dimension, but since they are of different type, the lemma still holds.

A.2. Omitted proofs from chapter 1

In this section, we restate and prove some results from chapter 1, that we left out for betterreadability.

Proposition A.13. Let E : Md → Md be a positive linear map. The following statements areequivalent:

(1) E(1) > 0

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(2) E : Cd → Cd (equivalence with the first item stated without proof in [1]).

(3) for X,Y > 0, YE(X.X†)Y† : Cd → Cd.

(4) E∗(σ) , 0 for all σ ≥ 0

Proof. ”(1)⇐ (2)” clear. ”(2)⇔ (3)” clear, since X,Y are invertible, hence for any A > 0, wehave A = X−1BX−† > 0 with B > 0 and YE(XAX†)Y† = YE(B)Y−1 > 0. ”(1)⇒ (2)” clear byusing the equivalence of (2) and (3) and noting that if there was an A > 0 s.th. E(A) ≥ 0 andsingular, then by decomposing A = XX† with X > 0 we obtain E′(.) := E(X.X†) fulfilling E′(1)is not positive definite, which is a contradiction. ”(2)⇔ (4)” Let ρ ∈ Cd be arbitrary. Then forall positive semidefinite σ we have by the above proposition tr(ρσ) > 0. tr(E(ρ)σ) > 0, too. Buttr(E(ρ)σ) = tr(ρE∗(σ)), which is always positive (by the above proposition, since ρ is positivedefinite) iff E∗(σ) is nonzero.

Lemma A.14 (noted without proof in [40]). Given A ≥ 0 s.th. supp(E(A)) ⊂ supp(A), thenE(PMdP) ⊂ PMdP, where P is the orthogonal projector onto the support of A.

Proof. From supp(E(A)) ⊂ supp(A) and P the orthogonal projector onto the support of A we getA = PAP, hence supp(E(PAP)) ⊂ supp(PAP). Now since PMdP consists of precisely thosematrices with support (and range) in supp P, we need to show that for any B ∈ B, supp(E(PBP)) ⊂supp(PAP).First assume B ∈ Cd. Now by construction, supp(PBP) ⊂ supp(PAP). Assume now thatsupp(E(PBP)) 1 supp(PAP). Hence we have a vector x < supp(PAP) s.th. E(PBP)x , 0.WLOG, we can choose x s.th. x†E(PBP)x > 0, for the following reason:Sine E(PBP) ≥ 0, all diagonal entries must be nonnegative. Furthermore, in the basis where P isdiagonal with P = diag(1, . . . , 1, 0, . . . , 0), PMdP are the matrices with entries only in the upperleft corner (dimension given by the rank of P). Now if E(PBP) has nonzero entries outside this

area, there must also be a positive diagonal entry, since matrices of the form(a bb 0

)have negative

determinant and are hence not positive semidefinite. If i is the index of the positive diagonalelement, x = (0, . . . , 0, xi, 0 . . . , 0)† < supp(PBP) fulfills the requirements.Since PAP ≥ 0 and PBP ≥ 0 fulfill supp(PBP) ⊂ supp(PAP) we can find ε > 0 s.th. PAP −εPBP ≥ 0 and thus E(PA − εBP) ≥ 0, but x†E(PA − εBP)x = −εx†E(PBP)x < 0, which isa contradiction to the positiveness of E. Hence supp(E(PBP)) ⊂ supp(E(PAP)) ⊂ supp(PAP),which was to be shown.Now for arbitrary B ∈ Md the claim follows by decomposing B into four positive semidefinitematrices, hence decomposing PBP into four positive semidefinite matrices in PMdP and usingthe above.

Lemma A.15 (noted without proof in [40]). E is irreducible iff there does not exist a nontrivialorthogonal projection P s.th. tr(E(P)(1 − P)) = 0.

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Proof. ”⇐”: Suppose E was reducible, i.e. E(PMdP) ⊂ PMdP for some orthogonal projectionP. Then especially supp(E(P)) ⊂ supp(P) and hence tr(E(P)(1 − P)) = 0.”⇒”: Let A ∈ Md be positive. Then tr(E(PAP)(1 − P)) ≥ 0 since the first part is a positivesemidefinite matrix by positivity of E and the second is a projector (same argument as above: Usespectral projections). Also, we observe:

tr(E(PAP)(1 − P)) = tr(APE∗(1 − P)P)

≤ ‖A‖∞ tr(P1PE∗(1 − P)) = ‖A‖∞ tr(E(P)(1 − P)) = 0

The inequality follows from the fact that PE(1 − P)P is still positive semi-definite, since it isHermitian (as the projectors are Hermitian) and since x†E(1 − P)x > 0 for all x ∈ Cd impliesalso x†PE(1 − P)Px > 0. Then, in the spectral basis of A =

∑i λiPi we have tr(APE∗(1 − P)P) =∑

i λi tr(PiPE∗(1 − P)P) ≤ ‖A‖∞ tr(PE∗(1 − P)P). Now tr(E(PAP)(1 − P)) = 0 implies thatsupp(E(PAP)) ⊂ supp(P). This in turn, via the above lemma, implies the claim.

Lemma A.16 ([40]). Irreducibility is invariant under similarity transformations, i.e. for C > 0E is irreducible iff E′(·) := CE(C−1(·)C−†)C† is irreducible.

Proof. We follow [40]. Assume Q , 0,1 and E(QMdQ) ⊂ QMdQ. Then if P is the supportprojection onto CQC†, we have that E′(PMdP) ⊂ PMdP. Since C is invertible, P,Q have thesame rank, i.e. P , 0,1 iff Q is, hence E is irreducible, iff E′ is.

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