ERGODIC PROPERTIES OF BOGOLIUBOV AUTOMORPHISMS IN FREE PROBABILITY

October 7, 2010 16:16 WSPC/S0219-0257 102-IDAQPRTS0219025710004140

Infinite Dimensional Analysis, Quantum Probabilityand Related TopicsVol. 13, No. 3 (2010) 393–411c© World Scientific Publishing CompanyDOI: 10.1142/S0219025710004140

ERGODIC PROPERTIES OF BOGOLIUBOVAUTOMORPHISMS IN FREE PROBABILITY

FRANCESCO FIDALEO∗ and FARRUKH MUKHAMEDOV†

Department of Computer and Theoretical Science,Faculty of Science, IIUM, P. O. Box 141,

25710 Kuantan, Pahang, Malaysia∗[email protected]

†[email protected]†farrukh [email protected]

Received 29 March 2010Communicated by F. Fagnola

We show that some C∗-dynamical systems obtained by free Fock quantization of classi-cal ones, enjoy ergodic properties much stronger than their boson or fermion analogous.Namely, if the classical dynamical system (X, T, µ) is ergodic but not weakly mixing,then the resulting free quantized system (G, α) is uniquely ergodic (w.r.t. the fixed pointalgebra) but not uniquely weak mixing. The same happens if we quantize a classical sys-tem (X, T, µ) which is weakly mixing but not mixing. In this case, the free quantizedsystem is uniquely weak mixing but not uniquely mixing. Finally, a free quantized sys-tem arising from a classical mixing dynamical system, will be uniquely mixing. In sucha way, it is possible to exhibit uniquely weak mixing and uniquely mixing C∗-dynamicalsystems whose Gelfand–Naimark–Segal representation associated to the unique invari-ant state generates a von Neumann factor of one of the following types: I∞, II 1, III λ

where λ ∈ (0, 1]. The resulting scenario is then quite different from the classical one. Infact, if a classical system is uniquely mixing, it is conjugate to the trivial one consist-ing of a singleton. For the sake of completeness, the results listed above are extended

to the q-Commutation Relations, provided |q| <√

2 − 1. The last result has a self-contained meaning as we prove that the involved C∗-dynamical systems based on theq-Commutation Relations are conjugate to the corresponding one arising from the freecase (i.e. q = 0), at least if |q| <

√2 − 1.

Keywords: Unique ergodicity; mixing; Bogoliubov automorphisms; C∗-dynamicalsystems; free probability.

AMS Subject Classification: 37A30, 46L55, 20E06

∗Permanent address: Dipartimento di Matematica, II Universita di Roma “Tor Vergata,” Via dellaRicerca Scientifica, 00133 Roma, Italia.

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http://dx.doi.org/10.1142/S0219025710004140


394 F. Fidaleo & F. Mukhamedov

1. Introduction

The study of quantum dynamical systems has had an impetuous growth in the lastyears, in view of natural applications to various field of mathematics and physics.It is then of interest to understand among the various ergodic properties, whichones survive and are meaningful by passing from the classical to the quantum case.Due to noncommutativity, the latter situation is much more complicated than theformer. The reader is referred e.g., to Refs. 3, 13, 14, 19 and 29 for further detailsrelative to some differences between the classical and the quantum situations. Itis therefore natural to study the possible generalizations to quantum case of thevarious ergodic properties known for classical dynamical systems.

By coming back to the classical case, one among the strongest ergodic propertiesenjoyed by a dynamical system (Ω, T ) consisting of a compact metric space Ω anda homeomorphism T , is the unique ergodicity. The last property means that thereexists a unique invariant Borel measure µ for T . It is known (cf. Ref. 24) that theunique ergodicity is equivalent to the uniform convergence, for each f ∈ C(Ω) ofthe Cesaro mean 1

n

∑n−1k=0 f T−k to

∫fdµ. A pivotal example of classical uniquely

ergodic dynamical system is given by an irrational rotation on the unit circle (seeRef. 9 for further examples).

A natural generalization of the unique ergodicity is formulated as follows, seee.g., Ref. 1. Let (A, α) be a C∗-dynamical system consisting of the unital C∗-algebraA and the automorphism α. It is called uniquely ergodic with respect to the fixedpoint algebra (cf. Refs. 1, 2 and 27) if the following limit exists in norm

limn→+∞

1n

n−1∑k=0

αn(a) = E(a), a ∈ A.

Here, E is nothing but the conditional expectation onto the fixed point algebra,provided such a conditional expectation exists. The previously mentioned classicalsituation corresponds to the case when A is Abelian (thus, α(f) = f T−1 for somehomeomorphism T : Ω → Ω of the spectrum Ω of A), and E = ω( · )1 for somestate ω on A. It is readily seen that ω is indeed the unique invariant state underthe action of the automorphism α. The cases when one replaces the automorphismα with a unital positive linear map are investigated in Refs. 2 and 15.a

The natural generalization of the unique ergodicity is the unique weak mixing,which means that

limn→+∞

1n

n−1∑k=0

|ϕ(αk(a)) − ϕ(E(a))| = 0, a ∈ A,

for every ϕ ∈ S(A). As before, E is the unique conditional expectation projectingonto the fixed point subalgebra. It was firstly introduced and studied in Ref. 27.

aNotice that when α is merely a unital positive linear map, the fixed point subspace is in generalnot a subalgebra, see Ref. 15, Sec. 2.

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Ergodic Properties 395

The reader is referred to Ref. 15 for the systematic investigation of the unique weakmixing for dynamical systems arising from free probability.

Finally, the strongest natural ergodic property, denoted as unique mixing, wasdefined and studied in Ref. 14. We simply require that

limn

ϕ(αn(a)) = ϕ(E(a)), a ∈ A, (1.1)

for every ϕ ∈ S(A).The property (1.1) of the convergence to the equilibrium is perfectly meaningful

in the quantum setting but its classical counterpart is the following: if a classicalsystem fulfills (1.1) with E(f) =

∫fdµ, the support of the unique invariant measure

µ is a singleton. This means that a uniquely mixing classical dynamical system isconjugate to the trivial one-point dynamical system. This is nothing but the contentof Sec. 4 of Ref. 14.

The noncommutative situation appears therefore much more interesting thanthe classical one. Indeed, it is possible to exhibit nontrivial examples of uniquelymixing C∗-dynamical system in the quantum setting, for which the fixed pointalgebra is trivial or nontrivial as well, see Ref. 14. Such examples are constructed byquantizing the shift on 2(Z) on the Boltzmann–Fock space F(2(Z)). The naturalgeneralization to the case when the fixed point algebra is nontrivial (i.e. when theinvariant state in not unique) arises from the shift on the amalgamated free productof C∗-algebras, or by considering length-preserving automorphisms on the reducedC∗-algebra of RD-groups (see Ref. 21 for the definition). The reader is referredalso to Refs. 1 and 15 for further details. Finally, in Ref. 11 it has been establishedthat the shift automorphism of the q-deformed Commutation Relations enjoys theunique mixing property as well. We can then exhibit uniquely mixing C∗-dynamicalsystems for which the von Neumann algebra generated by the GNS representationof the unique invariant state is a type I∞ factor or a type II 1 factor. The formeris obtained by considering the unital C∗-algebra R acting on the Boltzmann–Fockspace F(H), generated by all the annihilators a(f), f ∈ H, whereas the latterarises by considering the unital C∗-algebra G generated by the self-adjoint part ofthe annihilators a(f) + a+(f), f ∈ H.

It is then natural to address the possibility to exhibit further examples ofC∗-dynamical systems enjoying all the ergodic properties listed above, such that thevon Neumann algebra generated by the GNS representation of the unique invariantstate is a factor of different type from the previous ones. The aim of this paperis to show that this is indeed possible by quantizing classical dynamical systemssatisfying the corresponding ergodic properties. This simply means that ergodic,weakly mixing and mixing classical dynamical systems will lead to uniquely ergodic,uniquely weak mixing and uniquely mixing quantized dynamical systems, respec-tively.

Let H be a Hilbert space. Consider, for −1 ≤ q ≤ 1, the q-canonical commuta-tion relations for annihilators a(f) and creators a+(f):

a(f)a+(g) − qa+(g)a(f) = 〈g, f〉H1, f, g ∈ H. (1.2)

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The value q = −1 is the Fermi (Canonical Anticommutation Relation) case, whereasq = 1 is the Bosonic (Canonical Commutation Relation) case, and finally q = 0corresponds to the Boltzmann (or free) case. For unitaries U acting on H, theBogoliubov automorphisms (e.g., Ref. 6), uniquely defined as αU (a(f)) := a(Uf),are widely investigated in the Fermi and the Bose cases for the natural physi-cal applications. The reader is referred to Refs. 4, 8, 10, 28 and 32, for variousresults and applications, including the computation of various kind of entropies ofthe Bogoliubov automorphisms. In Ref. 7 (see also Ref. 35), it has been pointedout briefly that the free shift of the Cuntz algebra O∞ and consequently on thereduced C∗-algebra of the free group on infinitely many generators C∗

r (F∞), is“highly ergodic.” In this paper (see also Ref. 14) the meaning of the previous sen-tence seems to be clarified: all the free Bogoliubov automorphisms arising fromergodic, weakly mixing or mixing, including the shift, are either uniquely ergodicor uniquely weak mixing or uniquely mixing, respectively, that is they enjoy verystrong ergodic properties.

In order to carry out our plan, we start from a classical dynamical sys-tem (X, T, µ) consisting of a probability space (X, µ), and a measure preservinginvertible transformation T : X → X . By using the Shlyakhtenko construction onthe Boltzmann–Fock space (cf. Refs. 18 and 31), we consider the Bogoliubov auto-morphism αU given by αU (a(f)) = a(Uf), relative to the unitary U associated tothe measure-preserving transformation T . We obtain the following results. If theclassical dynamical system (X, T, µ) is ergodic but not weakly mixing, then theresulting free quantized system (G, α) always has a nontrivial fixed point algebra,and is uniquely ergodic but not uniquely weak mixing. If we quantize a classical sys-tem (X, T, µ) which is weakly mixing but not mixing, the resulting free quantizedsystem is uniquely weak mixing but not uniquely mixing. Finally, if we quantize amixing system (X, T, µ), the resulting quantum system will be uniquely mixing. Insuch a way, by tensoring with a multiplicity space on which the modular operator(see Sec. 4 below) is acting, it is possible to exhibit uniquely weak mixing but notuniquely mixing, and uniquely mixing C∗-dynamical systems whose von Neumannalgebra generated by the GNS representation associated to the unique invariantstate is a factor of all the types I∞, II 1 or III λ where λ ∈ (0, 1].

For the sake of completeness, our results are extended to the C∗-dynamicalsystems based on the q-commutation relations, provided |q| <

√2 − 1. It is well

known that if q is sufficiently small, the unital C∗-algebra Rq generated by all theannihilators aq(f), f ∈ H is isomorphic with the analogous one R correspondingto q = 0. The construction of such an isomorphism is rather involved, see e.g.,Refs. 12 and 23. The key point is to show that we can choose such an automorphismθq : Rq → R such that it intertwines the corresponding Bogoliubov automorphisms,see Theorem 5.3 below. Notice that the last result has a self-contained meaningas well.

To end the present introduction, we point out few things. At the classical levelwe have a wide class of uniquely ergodic dynamical systems, see e.g., Ref. 9 and

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the literature cited therein. In addition, starting from a measure-preserving ergodicdynamical system (X, T, µ), it is possible to construct in a canonical way a uniquelyergodic classical dynamical system (C(Y ), αS) such that (Y, S, ν) is conjugate to(X, T, µ), ν being the unique invariant probability measure on Y invariant underS.b This is nothing but the Jewett–Krieger theorem, see Refs. 20 and 25. Forthe intermediate weak mixing situation, nothing is yet known and the Jewett–Krieger theorem is not yet available in this case. Finally, for the mixing case, anyclassical dynamical system enjoying (1.1) with E = ω( · )1 is conjugate to the one-point trivial dynamical system and then the Jewett–Krieger theorem cannot becarried out. Notice that our approach is similar to the Jewett and Krieger oneat least in principle. Namely, starting from a classical dynamical system based onmeasure-preserving transformation which is ergodic, weakly mixing or mixing, wecan construct in a functorial way, nontrivial quantum dynamical systems, one foreach type I∞, II 1 or III λ, λ ∈ (0, 1] of von Neumann factor, enjoying the uniqueergodicity, unique weak mixing or unique mixing, respectively.

2. Preliminaries

Let A be a C∗-algebra with unit 1. Denote S(A) the set of all states on A. Fora (discrete) C∗-dynamical system we mean the pair (A, α) consisting of a unitalC∗-algebra and an automorphism α of A. The triplet (A, α, ω) consisting of aC∗-algebra, an automorphism as before, and finally a state ω ∈ S(A) invariantunder the action of α is called a C∗-dynamical system as well.

Consider the classical dynamical system (X, T, µ) based on a probability space(X, µ), and a measure-preserving invertible transformation T : X → X . It is wellknown that T induces a unitary transformation acting on L2(X, µ). Consider thenatural restriction of U to H := L2(X, µ) C1, 1 being the constant function. Itis known (cf. Refs. 16 and 24) that the dynamical system (X, T, µ) is ergodic, weakmixing, mixing iff

limn→∞

1n

n∑k=1

〈Ukξ, η〉 = 0, ξ, η ∈ H, (2.1)

limn→∞

1n

n∑k=1

|〈Ukξ, η〉| = 0, ξ, η ∈ H, (2.2)

limn→∞〈Unξ, η〉 = 0, ξ, η ∈ H, (2.3)

respectively. In what follows, we call for short, any unitary U acting on a Hilbertspace H, ergodic, weakly mixing, or mixing if it satisfies (2.1)–(2.3), respectively.

bThe measure-preserving transformations (Xj , Tj , µj), j = 1, 2, are said to be conjugate if thereexist µj -measurable sets Aj ∈ Xj of full measure such that Tj(Aj) = Aj , and a one-to-onemeasure-preserving map S : A1 → A2 such that T2 = S T1 S−1.

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We recall that the lower density D∗(A) of some A ⊂ N is defined by

D∗(A) := lim infn→∞

|A ∩ [0, n]|n + 1

,

see e.g., Chap. 3, of Ref. 16. The lower density of a subsequence kjj∈N of naturalnumbers means the lower density of the subset kj : j ∈ N ⊂ N.

For the convenience of the reader we report the following characterization ofthe weak mixing and mixing for a unitary. Indeed, a unitary U acting on a Hilbertspace H is weak mixing (resp. mixing) according to our definition, if and only iffor each x ∈ H, and for each subsequence kjj∈N of strictly positive lower density(resp. for each subsequence kjj∈N)

limn→∞

∥∥∥∥∥ 1n

n∑k=1

Ukj x

∥∥∥∥∥ = 0. (2.4)

For the generalization of the above characterizations of the weak mixing and mixingto power bounded operators acting on Banach spaces and for further details, werefer the reader to the original papers, Refs. 22 and 26, respectively.

Let (A, α) be a C∗-dynamical system, and E : A → A a linear map. Supposethat

limn→+∞

1n

n−1∑k=0

ϕ(αk(x)) = ϕ(E(x)), x ∈ A, ϕ ∈ S(A), (2.5)

limn→+∞

1n

n−1∑k=0

|ϕ(αk(x)) − ϕ(E(x))| = 0, x ∈ A, ϕ ∈ S(A), (2.6)

or finally,

limn→+∞ ϕ(αn(x)) = ϕ(E(x)), x ∈ A, ϕ ∈ S(A). (2.7)

It can readily be seen (cf. Refs. 1 and 15) that the map E is a conditional expectationprojecting onto the fixed point subalgebra Aα := x ∈ A : α(x) = x. Furthermore,E is invariant w.r.t. α.

Definition 2.1. (A, α) is said to be uniquely ergodic, uniquely weak mixing oruniquely mixing w.r.t. the fixed point subalgebra, if (2.5), (2.6) or (2.7) holds true,respectively.

If E = ω( · )1, then we simply call the dynamical system (A, α) uniquelyergodic, uniquely weak mixing or uniquely mixing (UE, UWM and UM for short),respectively.c

By using the Jordan decomposition of bounded linear functionals, one canreplace S(A) with A∗ everywhere in Definition 2.1. We refer to Refs. 33 and 34for standard results on the operator algebras and the modular theory.

cIf E = ω( · )1, then ω is the unique invariant state for α, see Ref. 1.

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Let H be a Hilbert space. The Boltzmann–Fock space (called also the full Fockspace) F(H) is defined by

F(H) := CΩ ⊕∞n=1 H⊗n.

The vector Ω is called the vacuum vector, and the vector state ω := 〈 ·Ω, Ω〉 thevacuum state. For f ∈ H, the left creator a+(f) acts on F(H) by

a+(f)Ω = f, a+(f)f1 ⊗ · · · ⊗ fn = f ⊗ f1 ⊗ · · · ⊗ fn, (2.8)

and its adjoint is the left annihilator a(f) given by

a(f)Ω = 0, a(f)f1 ⊗ · · · ⊗ fn = 〈f1, f〉f2 ⊗ · · · ⊗ fn.

Notice that the map f → a(f) is antilinear.d It is easily seen that a(f)∗ = a+(f),and the a(f) satisfy the commutation rule (1.2) with q = 0. Let P : H → H be acontraction. Then

F(P )f1 ⊗ · · · ⊗ fn := (Pf1) ⊗ · · · ⊗ (Pfn)

uniquely defines a contraction F(P ) : F(H) → F(H). We are interested in the casewhen U is unitary. In that case F(U) is unitary as well, and defines in a canonicalway an automorphism αU of the unital C∗-algebra generated by the annihilatorsa(f)|f ∈ H, as well as that generated by their self-adjoint part s(f)|f ∈ H,where

s(f) = a(f) + a+(f). (2.9)

On the generators, we simply get

αU (a(f)) = a(Uf), αU (s(f)) = s(Uf).

Such an αU is called the Bogoliubov automorphism induced by the unitary U . Thereader is referred to Refs. 5, 31 and 36 for the quantization of more general con-tractions, the generalization to the case of the q-Commutation Relations, and forfurther details. Notice that the dynamical and topological entropies of such kind ofautomorphisms were intensively studied in Refs. 4, 17, 32 and 35 for the CAR andthe CCR cases. The unique case relative to q = 0 concerns the quantization of theshift on 2(Z), see e.g., Refs. 7 and 8. Concerning the entropy, nothing is yet knownfor a general Bogoliubov automorphism in the case −1 < q < 1.

3. Ergodic Properties of Bogoliubov Automorphisms

In this paper we assume that all the Hilbert spaces we deal with are separable evenif most of the results can be extended to the non-separable case.

dIn this paper the inner product is linear in the first variable. In addition, the left creators andleft annihilators are denoted simply as creators and annihilators.

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Let H be a Hilbert space and U : H → H be a unitary. By αU we denote thecorresponding Bogoliubov automorphism as explained above. We start with thefollowing estimate crucial in the sequel.

Proposition 3.1. Let kll∈N be any subsequence of natural numbers, and U be aunitary operator. Under the above notations, we get

∥∥∥∥∥N∑

l=1

αkl

U (a+(f1) · · · a+(fm)a(g1) · · ·a(gn))

∥∥∥∥∥≤∥∥∥∥∥

N∑l=1

U−klf1 ⊗ · · · ⊗ U−klfm ⊗ Uklgn ⊗ · · · ⊗ Uklg1

∥∥∥∥∥ .

Proof. It is enough to consider x ∈ H⊗t with t ≥ n. By using any orthonormalbasis ejj∈J for H, we can symbolically write

x =∑

σ1,...,σn,s

xσ1,...,σn,seσ1 ⊗ · · · ⊗ eσn ⊗ ξs,

with 〈ξr, ξs〉 = δrs. We can also assume that the xσ1,...,σn,s are zero for finitely manyof them. Put F := f1 ⊗ · · · ⊗ fm, G := gn ⊗ · · · ⊗ g1. We have

Γ :=N∑

l=1

αkl

U (a+(f1) · · · a+(fm)a(g1) · · · a(gn))x

=N∑

l=1

∑s

⟨ ∑σ1,...,σn

xσ1,...,σn,seσ1 ⊗ · · · ⊗ eσn , (U⊗n)klG

⟩(U⊗m)klF ⊗ ξs.

It follows that 〈Γ, Γ〉 can be viewed as a linear combination of inner products ofelements of H⊗(2n+m). By using the Schwarz inequality and the fact that the flipoperator x ⊗ y ∈ H⊗H → y ⊗ x is unitary, we obtain

〈Γ, Γ〉 =∑s

⟨( ∑σ1,...,σn

xσ1,...,σn,seσ1 ⊗ · · · ⊗ eσn

)

⊗(

N∑l=1

(U⊗m)−klF ⊗ (U⊗n)klG

),

(N∑

l=1

(U⊗n)klG ⊗ (U⊗m)−klF

)

⊗( ∑

σ1,...,σn

xσ1,...,σn,seσ1 ⊗ · · · ⊗ eσn

)⟩

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≤∥∥∥∥∥

N∑l=1

(U⊗m)−kl ⊗ (U⊗n)klF ⊗ G

∥∥∥∥∥2

·∑s

∥∥∥∥∥∑

σ1,...,σn

xσ1,...,σn,seσ1 ⊗ · · · ⊗ eσn

∥∥∥∥∥2

=

∥∥∥∥∥N∑

l=1

(U⊗m)−kl ⊗ (U⊗n)klF ⊗ G

∥∥∥∥∥2

‖x‖2.

Let HR be a real Hilbert space, and UR : HR → HR be an orthogonal transformation.Extend UR linearly on the complex field to HC := HR + iHR, and denote such aunitary operator as UC. Denote T := z ∈ C | |z| = 1. We report the followingknown fact for the convenience of the reader.

Lemma 3.2. If σpp(UC) = ∅, then UR ⊗ UR has a nontrivial invariant vector.

Proof. Let eiθ ∈ σpp(UC) with eigenvector v = x + iy, with x, y ∈ HR. It followsthat, if eiθ = ±1, then UR has an eigenvector corresponding to eiθ. If eiθ ∈ T\±1,then UR has

( cos θ −sin θsin θ cos θ

)as a direct summand. Then

UR

(x

y

)=

(cos θ −sin θ

sin θ cos θ

)(x

y

).

The vector we are searching for is nothing but x ⊗ x + y ⊗ y.

Now consider a real subspace K ⊂ H of the Hilbert space H and suppose thatthe unitary operator U acting on H satisfies UK ⊂ K, U∗K ⊂ K. In this case,UK defines an orthogonal transformation on K, when the last is equipped withthe inner product (x, y) := Re〈x, y〉. Let (RK, αU ) be the C∗-dynamical system,where RK is the unital C∗-algebra acting on F(H) generated by a(f) | f ∈ K,and αU the restriction of the Bogoliubov automorphism generated by U to RK. TheC∗-dynamical system (GK, αU ) consists of the unital C∗-algebra acting on F(H)generated by s(f) | f ∈ K and the restriction of the Bogoliubov automorphismgenerated by U to GK. Here, s(f) is given in (2.9). Note that this is a VoiculescuC∗-Gaussian functor, see Ref. 36. When K ⊂ H is fixed in the sequel, we simplywrite (R, α) and (G, α), respectively.

Proposition 3.3. If U on K is ergodic, then the dynamical systems (R, α), (G, α)are uniquely ergodic w.r.t. the fixed point algebra.

Proof. By a standard approximation argument, it is enough to consider the casewhen

A := a+(f1) · · ·a+(fm)a(g1) · · · a(gn), (3.1)

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where the f1, . . . , fm, g1 . . . , gn are either eigenvectors of U , or belong to the Hilbertsubspace relative to the continuous spectrum of U . If f1, . . . , fm, g1 . . . , gn areeigenvectors of U with corresponding eigenvalues eiθ1 , . . . , eiθm , eiΘ1 , . . . , eiΘn suchthat

m∑i=1

θi −n∑

i=1

Θi = 2hπ

for some integer h ∈ Z, we conclude that A is invariant under α. In all the remainingcases, the vector f1 ⊗ · · · ⊗ fm ⊗ g1 ⊗ · · · ⊗ gn ∈ H⊗(m+n) is not invariant for theunitary (U∗)⊗m ⊗ U⊗n. Then by Proposition 3.1 and Mean Ergodic theorem, weget in this situation, limN

1N

∑Nk=1 αk(A) = 0, and the proof follows.

Proposition 3.4. If U on K is weakly mixing (resp. mixing), then the dynamicalsystems (R, α), (G, α) are UWM (resp. UM ) with the vacuum state the uniqueinvariant state under α.

Proof. Let A be as in (3.1). We have for any subsequence kll∈N of naturalnumbers of positive lower density, limN

1N

∑Nl=1 αkl(A) = 0 by taking into account

Proposition 3.1 and the fact that U , and then (U∗)⊗m ⊗ U⊗n, is weakly mixing,see (2.4) or the original paper Ref. 22. Again by (2.4), this implies that, for eachX ∈ R such that ω(X) = 0, the sequence αn(X)n∈N is (uniformly) weakly mixingat 0. It turns out to be equivalent to the fact that (R, α) is UWM with ω the uniqueinvariant state. In the mixing case, by Proposition 3.1, we have for operators A asbefore and any subsequence kll∈N of natural numbers, limN

1N

∑Nl=1 αkl(A) = 0,

see (2.4) or the original paper, Ref. 26. The proof follows by Proposition 2.3 ofRef. 14.

Now we show that the quantized systems arising from ergodic but not weaklymixing classical dynamical systems, cannot be UWM w.r.t. the fixed algebra. Thesame will happen in the weak mixing situation: the resulting quantum system can-not be UM.

Proposition 3.5. Let U be ergodic (resp. weakly mixing) and suppose that thereexists some f ∈ K such that the sequence Ukfk∈N is not weakly mixing (resp.mixing) at 0. Then the dynamical systems (R, α), (G, α) cannot be UWM (resp.UM ) w.r.t. the fixed point algebra.

Proof. Let f ∈ K such that Ukfk∈N is not weakly mixing (resp. mixing) at 0.By Proposition 3.3, the dynamical systems (R, α), (G, α) are UE w.r.t. the fixedpoint algebra. Thus,

limN

1N

N∑k=1

αk(a+(f)) = E(a+(f)) = 0 = EG(s(f)),

E being the conditional expectation onto Rα. According to (2.4), there exists a sub-sequence kll∈N of natural numbers of positive lower density (resp. a subsequence

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of natural numbers) such that

lim supN

∥∥∥∥∥ 1N

N∑l=1

Uklf

∥∥∥∥∥ > 0.

Suppose that (R, α) or (G, α) is UWM (resp. UM) w.r.t. the fixed point algebra.Then one gets

0 = limN

1N

∥∥∥∥∥N∑

k=1

αk(s(f))Ω

∥∥∥∥∥ = limN

∥∥∥∥∥ 1N

N∑k=1

αk(a+(f))Ω

∥∥∥∥∥= lim sup

N

∥∥∥∥∥ 1N

N∑l=1

Uklf

∥∥∥∥∥ > 0

which is a contradiction.

We end the present section with the following comment. In the case of the CARalgebra CAR(L2(Rn)) on L2(Rn), it is straightforwardly proved (see e.g., p. 45,Ref. 6) that the C∗-dynamical system (CAR(L2(Rn)), τx, ω) is mixing for eachstate ω on the CAR, invariant w.r.t. the translations τx | x ∈ Rn. In the CARcase, after starting by a group of Bogoliubov automorphisms generated by a mixingunitary group (i.e. the translations), we obtain a quantized C∗-dynamical systemwhich is mixing. The result proved above for the free case is much more strong.Indeed, starting from any mixing unitary, the free quantized resulting system isuniquely mixing. In particular, there exists a unique invariant state, which is thevacuum. In the CAR case, it is well known that there are plenty of translationinvariant states.e

4. On the Type of the Factors Generated by theBogoliubov Automorphisms

In this section we construct C∗-dynamical systems enjoying the strong ergodicproperties listed in Sec. 2, and whose GNS representation relative to the Fockvacuum (which is the unique invariant state for the discrete dynamics in the caseof UWM and UM) generates type II 1 and type III λ, λ ∈ (0, 1], von Neumannfactors. This is done by quantizing any classical ergodic, weakly mixing or mixingdynamical system on the Boltzmann–Fock space.

eBy using the language in Sec. 5 of Ref. 14, this means that (CAR(L2(Rn)), τx) is (Fπω , ω)-mixingfor each translation invariant state ω. In the free case, (A, α) is (A∗, ω)-mixing, where α is theBogoliubov automorphism generated by any mixing unitary, and ω is the vacuum state. In theformer case, we have the convergence to the equilibrium only for the states in the folium Fπω

generated by the state ω (i.e. the folium made of all the functionals which are normal w.r.t. ω).In the latter case, the convergence to the equilibrium is for every functional in the dual A∗.

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Let (X, T, µ) be a classical dynamical system made of a probability space (X, µ),and a measure-preserving invertible transformation T : X → X . We assume thatL2(X, µ) is infinite-dimensional. Let

KR := (L2R(X, µ) R1)

⊗(⊕λ∈GR2). (4.1)

Here, 1 ∈ L2R(X, µ) is the constant function which is invariant under the action of

T , and G is any countable multiplicative subgroup of R+. Let uf := f T−1 and

v(t) :=⊕λ∈G

(cos(t ln λ) −sin(t ln λ)

sin(t ln λ) cos(t ln λ)

).

Then u ⊗ I and I ⊗ v(t) are orthogonal transformations acting on the real Hilbertspace KR satisfying [u ⊗ I, I ⊗ v(t)] = 0. Let KC be the complexification of KR

together with the positive non-singular generator A of the complexification of I ⊗v(t), as I ⊗ v(t) = I ⊗ ait =: Ait. Let H be the completion of KC with respect tothe inner product induced by A

〈x, y〉 := (2A(I + A)−1x, y), (4.2)

where ( · , · ) is the inner product of KC. Denote by U and V (t) the unitary extensionof the corresponding orthogonal operators to the whole H. Let F(H) be the fullFock space on H, together with the Fock vacuum vector Ω. Consider the unitalC∗-algebra G acting on F(H), generated by s(f) : | f ∈ KR where s(f) is givenin (2.9). Notice that Ω is cyclic for G and G′, that is Ω is a standard vector for G′′.The dynamical system under consideration is (G, α), where α is the automorphismon G induced by α(s(f)) := s(Uf). We refer the reader to Refs. 18 and 31 forfurther details.

Proposition 4.1. If (X, T, µ) is ergodic but not weakly mixing, then the fixed pointalgebra Gα is nontrivial.

Proof. As u is nontrivial and not weakly mixing, U has at least an eigenvalue χ inT\1. If χ = −1 there is a corresponding eigenvector f ∈ KR. An invariant elementunder the action of α is s(f)2. If χ ∈ T\±1, with the corresponding eigenvectorv = f + ig, then by Lemma 3.2, an invariant element is given by s(f)2 + s(g)2.

The main results of this paper are summarized in the following theorems.

Theorem 4.2. Let α be the Bogoliubov automorphism in G induced by α(s(f)) :=s(Uf). Then the following assertions hold true.

(i) If (X, T, µ) is ergodic but not weakly mixing, then (G, α) is UE but not UWMw.r.t. the fixed point algebra Gα, which is always nontrivial.

(ii) If (X, T, µ) is weakly mixing but not mixing, then (G, α) is UWM but not UM,

with ω the unique invariant state.(iii) If (X, T, µ) is mixing, then (G, α) is UM, with ω the unique invariant state.

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Proof. We start by noticing that if U is ergodic (resp. weakly mixing or mixing)in KC, then its extension on H is ergodic (resp. weakly mixing or mixing) as well.This easily follows by (4.2) as, for any subsequence kll∈N of natural number, weget ∥∥∥∥∥ 1

N

N∑l=1

Uklf

∥∥∥∥∥H

≤ ‖A‖1/2B(KC)

∥∥∥∥∥ 1N

N∑k=1

Ukf

∥∥∥∥∥KC

. (4.3)

Then by Proposition 3.3 (resp. Proposition 3.4), (G, α) is UE w.r.t. the fixed pointalgebra (resp. UWM or UM with the Fock vacuum ω as the unique invariant state).On the other hand, if u is not weakly mixing its pure point spectrum, and thenthat of U = u ⊗ I is nonvoid. Therefore the fixed point algebra Gα is nontrivialby Proposition 4.1. Let F now be a non-null function on L2(X, µ) with

∫Fdµ = 0

such that

lim supN

∥∥∥∥∥ 1N

N∑l=1

F T−kl

∥∥∥∥∥ > 0 (4.4)

for some subsequence kll∈N of natural numbers of positive lower density (resp. asubsequence of natural numbers). Notice that if G = G1 + iG2, then

∫ |G|2 dµ =∫(G2

1+G22)dµ. This means that (4.4) should be fulfilled at least by one of the real or

imaginary parts of F . Thus, we can assume without loss of generality, that F itselfis real. After choosing f := F ⊗ ξ with ξ ∈ ⊕λ∈GR2\0, we get

lim supN

∥∥∥∥∥ 1N

N∑l=1

Uklf

∥∥∥∥∥H

> 0

for the same subsequence kll∈N of natural numbers of positive lower density(resp. a subsequence of natural numbers) as before. Therefore, we conclude byProposition 3.5 that if (X, T, µ) is ergodic but not weakly mixing (resp. weaklymixing but not mixing), (G, α) cannot be UWM w.r.t. the fixed point algebra(resp. UM).

Notice that the GNS representation πω of G relative to the vacuum state ω,coincides up to unitary equivalence, with the defining representation of G itself onthe full Fock space F(H).

Theorem 4.3. For the C∗-dynamical systems considered above, we have that G′′ ∼=πω(G)′′ is a non-injective von Neumann factor of types II 1, III λ, λ ∈ (0, 1) or III 1,

whenever G is 1, λn : n = 0, 1, 2, . . . or Q+ respectively.

Proof. As we are assuming that L2(X, µ) is infinite-dimensional, we have thatthe positive operator A in (4.2), which is almost periodic in our construction (cf.Ref. 31), always has infinitely many mutually orthogonal eigenvectors correspondingto the eigenvalue 1. Let 1 = λ1 = λ2 = · · · = λN = · · · be an infinite sequence of

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such eigenvalues. We have 1√N

∑Nk=1

2√λk+

√λ−1

k

> 4 whenever N > 16. Therefore,

G′′ is not injective by Theorem 2.2 of Ref. 18. On the other hand, by Theorem 3.2of Ref. 18, the centralizer (G′′)ω has trivial relative commutant in G′′. This impliesthat G′′ is a factor. Finally, Theorem 3.3 of Ref. 18 (see also Ref. 31) provides theresult relative to the type of the factor G′′.

Notice that, with R the unital C∗-algebra generated by a(f) | f ∈ KR, R′′ ∼=πω(R)′′ is a type I∞ von Neumann factor, see e.g., Ref. 11 for the proof.

5. The Case of q-Commutation Relations

The present section is devoted to show that all the previous results can be extendedto the q-deformed commutation relations, at least for sufficiently small q.

Suppose −1 < q < 1. The concrete C∗-algebras Rq and its subalgebra Gq act onthe q-deformed Fock space Fq(H), which is the completion of the algebraic linearspan of the vacuum vector Ω, together with vectors

f1 ⊗ · · · ⊗ fn, fj ∈ H, j = 1, . . . , n, n = 1, 2, . . .

w.r.t. the q-deformed inner product

〈f1 ⊗ · · · ⊗ fn, g1 ⊗ · · · ⊗ gm〉q := δn,m

∑π∈Pn

qi(π)〈f1, gπ(1)〉 · · · 〈fn, gπ(n)〉, (5.1)

Pn being the symmetric group of n elements, and i(π) the number of inversions ofπ ∈ Pn. The creator a+

q (f) is defined as in (2.8), and the corresponding annihilatoris defined as

aq(f)Ω = 0,

aq(f)(f1 ⊗ · · · ⊗ fn) =n∑

k=1

qk−1〈fk, f〉f1 ⊗ · · · fk−1 ⊗ fk+1 ⊗ · · · ⊗ fn.

a+q (f) and aq(f) are adjoint to each other w.r.t. the inner product (5.1) and satisfy

the commutation relations (1.2). The Fock vacuum is defined as ωq := 〈 ·Ω, Ω〉.As for the free situation, a unitary acting on H induces a Bogoliubov automor-

phism αU on the unital C∗-algebra generated by all the annihilators aq(f), uniquelydefined as αq,U (aq(f)) = aq(Uf). Such an automorphism is unitarily implementedby the quantization Fq(U) given by

Fq(U)f1 ⊗ · · · ⊗ fn = (Uf1) ⊗ · · · ⊗ (Ufn).

To simplify, we drop U from the subscript in αq,U . The reader is referred to Ref. 5and the literature cited therein, for further details.

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Let (Gq, αq) be the C∗-dynamical system where Gq is the unital C∗-algebraacting on Fq(H) generated by sq(f)|f ∈ KR, with

sq(f) := aq(f) + a+q (f)

and then αq(sq(f)) = sq(Uf). Here, KR is given in (4.1), H is the completion ofKR + iKR w.r.t. the inner product given in (4.2), and finally U is the unitary actingon H as described in Sec. 4.

In order to extend our previous results to the q-Commutation Relations, weneed some preparatory results.

Proposition 5.1. Let Unn∈N and U be unitaries acting on H, together with thecorresponding Bogoliubov automorphisms α(n)

q n∈N and αq on Rq, respectively.If limn Un = U in the strong operator topology of B(H), then α

(n)q converges

pointwise in norm to αq.

Proof. It is enough to prove the assertion for each aq(f), f ∈ H. We get byRemark 1.2 of Ref. 5,

‖α(n)q (aq(f)) − αq(aq(f))‖B(Fq(H)) = ‖aq(Unf) − aq(Uf)‖B(Fq(H))

= ‖aq(Unf − Uf)‖B(Fq(H))

≤ (1/√

1 − |q|)‖(Unf − Uf)‖H.

Let R be any finite-dimensional Hilbert space whose dimension is equal to d,together with an orthonormal basis ejd

j=1. It is shown in Refs. 23 and 12 that,if |q| <

√2 − 1, the unital C∗-algebra Rq is isomorphic to R0 ≡ R via a map θ

sending aq(ej) to a0(ej)R. Here, R ∈ R0 is a positive element satisfying

R2 =d∑

j=1

a+0 (ej)a0(ej) +

d∑j,k=1

(a0(ej)Ra0(ek))∗(a0(ek)Ra0(ej)).

Let M :=∑d

j=1 a+q (ej)aq(ej). It is a positive operator. Furthermore, consider

the unitary operator

V =∞⊕

m=0

Vm : Fq(R) → F0(R)

defined recursively as

V0 := J0, Vm := (J1 ⊗ Vm−1)M1/2R⊗m , m = 1, 2, . . . ,

where J0, J1 are the identifications of the one-dimensional subspace spanned bythe vacuum vector and R in Fq(R), with the corresponding objects in F0(R),respectively. Then R can be written as R = V M1/2V ∗.

Lemma 5.2. Let U be a unitary acting on the finite-dimensional Hilbert space R.Then F0(U)R = RF0(U).

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Proof. By taking into account the definition of M1/2, V and R, it is enough toshow that M commutes with Fq(U). We get

Fq(U)Mf1 ⊗ · · · ⊗ fn

= Fq(U)n∑

i=1

qi−1fi ⊗ f1 ⊗ · · · ⊗ fi−1 ⊗ fi+1 ⊗ · · · ⊗ fn

=n∑

i=1

qi−1(Ufi) ⊗ (Uf1) ⊗ · · · ⊗ (Ufi−1) ⊗ (Ufi+1) ⊗ · · · ⊗ (Ufn)

= MFq(U)f1 ⊗ · · · ⊗ fn.

In addition, it is shown in Sec. 5 of Ref. 18 that the previous result extends to thecase of any separable Hilbert space R, where θ is the inductive limit of the corre-sponding isomorphisms θn for each increasing sequence of dn-dimensional subspacesRn such that

⋃n Rn is dense in R. We refer the reader to the above-mentioned

paper, Ref. 18 for further details relative to the isomorphism θ realizing the equiv-alence between Rq and R0, when |q| <

√2 − 1.f

Theorem 5.3. There exists an isomorphism θ : Rq → R0 which intertwines anyBogoliubov automorphism: θ αq = α0 θ, provided |q| <

√2 − 1.

Proof. Let U be the unitary acting on H generating the Bogoliubov automorphismon the algebras R0 and Rq, |q| <

√2 − 1. Let K be the Cayley transform of U ,

together with the resolution of the identity λ → E(λ) of K, which is supposed to beright-continuous (in the strong operator topology). Define the sawtooth function

h(λ) := λ − 2kπ, λ ∈ (2kπ, 2(k + 1)π], k ∈ Z.

It is easy to show that H :=∫

h(λ)dE(λ) is a bounded self-adjoint operator suchthat U = eiH . Fix an increasing sequence Hn of finite-dimensional subspaces suchthat

⋃n Hn is dense in H, together with the associated self-adjoint projections

Pn. Define Un := eiPnHPn . By Theorems VIII.25 and VIII.21 of Ref. 30, we getlimn Un = U in the strong operator topology of B(R). There exists an isomorphismθ : Rq → R0 intertwining α

(n)p and α

(n)0 for each n. This can be done by considering

for each fixed n, the sequence of unitary operators UnHm , m > n. Lemma 5.6 ofRef. 18 leads to the claim by taking into account Lemma 5.2. Fix A ∈ Rq. ByProposition 5.1 we get

θ(αq(A)) = θ(limn

α(n)q (A)

)= lim

nθ(α(n)

q (A)) = limn

α(n)0 (θ(A)) = α0(θ(A)),

which is the assertion.

Theorem 5.4. For the C∗-dynamical system (Gq, αq), all the assertions ofTheorem 4.2 hold true, provided |q| <

√2 − 1.

fSuch C∗-algebras are known in the literature as Eq(R) and E0(R), respectively.

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Proof. Fix the C∗-dynamical system (Rq, αq). By Theorem 5.3, it is conjugateto the C∗-dynamical system (R0, α0), where αq, α0 are Bogoliubov automorphismsgenerated on Rq, R0 by the same orthogonal operator. In addition, the isomorphismdescribed in Proposition 5.3 intertwines the corresponding Fock vacua ωq and ω0.g

By taking into account Proposition 3.1 and (4.3), (Rq, αq) is UE w.r.t. the fixedpoint subalgebra (resp. UWM or UM), provided U is ergodic (resp. weakly mixingor mixing). Then (Gq, αq) is by restriction, UE w.r.t. the fixed point subalgebra,UWM or UM, provided U is ergodic, weakly mixing or mixing, respectively. Onthe other hand, assume that the classical system (X, T, µ) is ergodic (resp. weaklymixing) but not weakly mixing (resp. mixing). We can choose as in the proof ofTheorem 4.2 f ∈ KR such that

lim supN

∥∥∥∥∥ 1N

N∑l=1

Uklf

∥∥∥∥∥H

> 0

for some subsequence kll∈N of natural numbers of positive lower density (resp. asubsequence of natural numbers). Then we have

0 = limN

1N

∥∥∥∥∥N∑

k=1

αkq (sq(f))Ω

∥∥∥∥∥= lim

N

∥∥∥∥∥ 1N

N∑k=1

αkq (a+

q (f))Ω

∥∥∥∥∥= lim sup

N

∥∥∥∥∥ 1N

N∑l=1

Uklf

∥∥∥∥∥ > 0

which is a contradiction. Thus, (Gq, αq) cannot be UWM w.r.t. the fixed pointsubalgebra (resp. UM). Finally, in the case when (X, T, µ) is weakly mixing ormixing, the fixed point algebra of (Rq, αq) is trivial. Then by restriction, the fixedpoint algebra of (Gq, αq) is trivial as well. In the ergodic case, the fixed point algebraof (Gq, αq) is nontrivial by Proposition 4.1.

Concerning the type of the factor generated by πωq(Gq)′′ ∼= G′′q , the same proof

as that in Theorem 4.3 allows us to say that G′′q is a non-injective von Neumann

factor of types II 1, III λ, λ ∈ (0, 1) or III 1, whenever G is 1, λn : n = 0, 1, 2, . . .or Q+ respectively. This holds true without any restriction on the value of q.

Consider now the q-shift. Namely, let u : 2R(Z) → 2

R(Z) be the shift acting on

2R(Z), and

KR := 2R(Z)

⊗(⊕λ∈G

R2

). (5.2)

gNotice that it is unclear if such isomorphism sends Gq onto G0. Thus, it is unclear if (Gq , αq) isconjugate to (G0, α0).

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As usual, H will be the completion of KR + iKR w.r.t. the inner product givenin (4.2). Define U and V (t) as in Sec. 4, and finally Gq as at the beginning ofthe present section. Also in this case G′′

q is a non-injective factor whose type isdetermined by the group G appearing in (5.2).

Recently, it was proven for each −1 < q < 1, that the q-shift is UM in the casewhen the modular theory is trivial (i.e. when in (5.2) G = 1). For the q-shift, thesame proof of Theorem 3 of Ref. 11 allows us to extend the previous results to allcases −1 < q < 1. We have then proven the following:

Proposition 5.5. For each −1 < q < 1, the C∗-dynamical system (Gq, αq) (αq

being the q-shift) is UM, with the Fock vacuum ωq the unique invariant state.

We might conjecture that all the results described in the present section for theC∗-dynamical systems based on the q-Commutation Relations, can be extended toall q ∈ (−1, 1). Unfortunately, it is not known if all the C∗-algebras Rq are isomor-phic for any q ∈ (−1, 1). In addition, estimates similar to that in Proposition 3.1or in Proposition 2 of Ref. 11 are not yet available for the general case when q = 0,and/or the involved Bogoliubov automorphism is not the shift.

Acknowledgment

F.M. thanks the MOHE grant FRGS0308-91.

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