Roots for quantum ergodic theory invon Neumann's work

Vol. 55 (2005) REPORTS ON MATHEMATICAL PHYSICS No. 1

ROOTS FOR QUANTUM ERGODIC THEORY IN VON NEUMANN'S WORK*

HEIDE NARNHOFER

Institut ftir Theoretische Physik, Universitiit Wien Boltzmanngasse 5, A-1090 Wien, Austria

(e-mail: narnh @ ap.univie.ac.at)

(Received November 4, 2004)

In ergodic theory von Neumann emphasized the spectral analysis of the unitary im- plementor and the possibility to express point translations as automorphisms over abelian algebras. Replacing the abelian algebras by noncommutative algebras good ergodic behaviour asks for type II and III algebras. The possibility for existing K-systems and Anosov systems in this framework is discussed. Von Neumanns example of a type III algebra is examined from this viewpoint.

PACS numbers: 03.67Hk, 05.30Fk. Keywords: von Neumann algebras, spectrum, defect indices, quantum ergodic theory, Anosov relations.

1. von Neumanns contribution to nonabelian algebras and ergodic theory

In the first part of the 20 th century two main mathematical disciplines started without referring one to another. On the one hand the results of Heisenberg made it necessary to understand and study noncommutative algebras. On the other hand, again influenced by physics, one wanted to control mixing properties and convergence properties of a deterministic dynamics in a way that justifies statistical mechanics. In both disciplines, in the theory of noncommutative algebras as well as in ergodic theory we owe major contributions to John von Neumann [1].

In the theory of noncommutative algebras von Neumann's motivation came from quantum mechanics. He realized that he could offer to Heisenberg's matrix mechanics the unique realization as operators in an infinite-dimensional Hilbert space [2, 6]. In a series of papers he studied the properties of operators. We owe to him the spectral theorem [4]. He realized in a precise analysis of unbounded operators, how much they depend on the domain on which they are defined, together with the possibility, that an unbounded operator can be Hermitian but not self-adjoint so that its functions cannot be defined over its spectrum.

*Lecture given at the von Neumann Centennial Conference, Budapest, October 15-20, 2003.

[93]

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Continuing the study of operators he realized that, if we concentrate on subsystems in the physical context we have, in the mathematical formulation, to concentrate on subalgebras [12, p. 10]. As a next step he evaluated the most appropriate topology on these subalgebras, namely the weak topology: the closure with respect to this topology coincides with a closure in an algebraic understanding, the algebra is its own bicommutant.

As a next task he examined together with F. J. Murray what kind of additional structural properties might characterize these subalgebras. Based on the spectral theorem they showed that the algebras can be considered as built by projections. They developed the definition of a dimension of these projections with respect to the algebra. In this way they could distinguish between three different types of von Neumann algebras. Of course von Neumann was not satisfied with the pure conceptual possibility of such different types but constructed examples to every of these types. The next problem, namely how far these examples are typical, i.e. whether under some appropriate restrictions all other examples are isomorphic to the one found by von Neu- mann, was attacked by von Neumann, but found its solution only much later [17, 18].

Maybe a bit unfortunate for applications in physics the construction of these factor II and factor III algebras let to examples which physicists could not recognize in their theories. Nevertheless the construction was inspired by a physical example: The Heisenberg algebra in its description as Weyl algebra can be looked at as being built by the continuous function on x together with the implementors of the automorphism group of the translation. This Weyl algebra has a unique representation as was shown by von Neumann in [6]. To obtain something new he manipulated the automorphism group, passing from a continuous group to a discrete group, and this manipulation found its counterpart that he moved from projections with discrete dimension to projections with continuous dimension [12]. An additional noncommutativity in the automorphism group let him go over to the type III algebra [15]. These manipulations in the explicit examples made the physicists feel that they are lead away from physical problems. Nevertheless von Neumann had the confidence that at least type II algebras should be of physical importance [12, p. 11]: "We think however that our results indicate that there is more point in assigning this role to II1. There are two chief reasons for this assertion: the existence of a trace and the behaviour of unbounded operators." This confidence seems to have been based on the beauty he saw in the type II algebras. As late as 1963 H. Araki [19] expressed his confidence that other than type I algebras appear in physical problems. The corresponding proof was given in [20].

Now we know that in contrast to von Neumann's believe the type III algebras are those that play an important role in physics: they appear in the thermodynamic limit of nonrelativistic particles at finite temperature [20]. The local algebras in quantum field theory, the subsystems as emphasized by von Neumann [21], are even in the ground state algebras of type III. In his article "On infinite direct products" [14] von Neumann already studied the example corresponding to infinite lattice systems that appear in physics. But at that time the relation to physics was not realized. Many-particle physics was done in Fock space and the concept of local subsystems

ROOTS FOR QUANTUM ERGODIC THEORY IN VON NEUMANN'S WORK 95

was not developed. Also von Neumann did not succeed to see the type III structure in his example and even made a wrong guess [14, p. 399]: "But we are rather inclined to surmise, that the above algebra will not be a factor of class (III)...". In 1940 von Neumann corrected this guess in [15] and remarked in the introduction that the algebras are type III. However, he did not give the proof in detail. It appeared in [22] ("..this is known but not published..", p. 587).

The excursion of von Neumann to ergodic theory was, compared to his contributions to noncommutative algebras, which lasted from 1927 to 1944, a short trip. Here we can concentrate mainly on three articles, "Proof of the quasiergodic hypothesis" [7], "Dynamical systems of continuous spectra" together with B. O. Koopman [8] and "Zur Operatormethode in der klassischen Mechanik" [10].

It was Koopman who realized that a measure space serves to construct a Hilbert space in which a measure preserving transformation o.n the measure space can be represented by a unitary operator. Evidently this offered an example to von Neumann, on which he could demonstrate the power of his spectral analysis [4]. Ergodicity becomes a consequence of the spectral properties of the unitary operator. Apart from the mathematical success he emphasized the role of ergodic theory "which plays so important a role in the foundation of classical statistical mechanics" [11, p. 274]. We have to notice that there is no remark so far how the ideas might be transported to quantum mechanics.

In his paper "Zur Operatorenmethode in der klassischen Mechanik" [10] he starts with praising this metlfod of translating problems into the language of operators and offering this way new possibilities for solutions. He argues as already in [7, p. 271] "which has an immediate physical consequence", that the measurable functions are those that admit physical interpretation in the sense that one can decide "physikalisch" by experiment, whether something belongs to a measurable set. In "Ueber messbare Abbildungen" [9] he shows the equivalence to talk about measure preserving "Punktabbildungen", transformations of points in a measure space, or to talk about automorphisms on the measurable functions over a measure space.

From this result it becomes evident how to formulate ergodic theory in the framework of quantum mechanics: The abelian von Neumann algebra over the measure space has to be replaced by a noncommutative von Neumann algebra. The dynamics can be realized as an automorphism group over the von Neumann algebra. Now the question arises whether automorphism groups can be represented by unitary operators and how far spectral properties reflect ergodic behaviour in the noncommutative case.

In the time when von Neumann was concerned with problems of ergodic theory the time was evidently not ready to treat quantum ergodic theory. Von Neumann himself had proven that quantum mechanics of finitely many particles had a unique representation as irreducible algebra over a Hilbert space [6]. Dynamics therefore reduced to the spectral theory of a Hamiltonian. A unique invariant state, the starting point in classical ergodic theory, corresponds to a unique nondegenerate eigenvalue of the Hamiltonian. But this cannot be transferred into the appropriate behaviour that time invariant means of operators are just c-numbers. On the contrary the in-

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variant mean of operators form the commutant of the unitary which implements the dynamics. If the Hamiltonian is nondegenerate, then this commutant becomes a maximally abelian algebra. Therefore mixing properties of some dynamics can only appear if the dynamics is not inner, and this can only happen if we consider factors of type II and type III, In fact, also von Neumann was interested in the possibility of not inner automorphisms together with their unitary implementation [16]. But at that time his interest came from the desire to restrict the class of von Neumann algebras up to isomorphism as much as possible. The automorphisms, respectively isomorphisms should serve as tool in this classification. In fact, the classification of [17] in the 70-ties was also based on automorphism properties, But with respect to ergodic properties in the framework of noncommutative algebras no hints could be found in von Neumann's articles. Nevertheless we will see that the fundamental results in quantum ergodic .theory need the classification of von Neumann algebras, their relation one to another, also the sensitivity of unbounded operators to domain questions. Even in the explicit examples that von Neumann constructed we can read off features of quantum ergodic theory.

2. Preliminary considerations on ergodicty in quantum theory

In the abelian situation von Neumann had shown that dynamics can be represented by a unitary operator in an appropriate Hilbert space. The constant measure defines the only invariant state if and only if the u'hitary has a unique discrete eigenvalue. This statement can immediately be translated to the quantum situation. But if we consider the observable algebra to be given by all bounded operators in a Hilbert space, as it is suggested by von Neumann's result on the Weyl algebra, then this is the end of any kind of ergodic theory: the ergodic hypothesis asks that the invariant mean of any operator, taken in the weak topology, has to be the expectation value in the unique invariant state. However, the functions of a unitary operator are invariant in time and violate the ergodic hypothesis. As a consequence no attempts were made to create a quantum ergodic theory till in the early 60ties Araki and Woods [20] recognized that the algebra of free bosons in thermal states is not a type I algebra. The tool was to prove that the time automorphism is not inner. Finer classification followed [23, 24]. Later Powers [25] showed that the algebras appearing in lattice theory are type III and are not all equivalent. In physics they are realized in the BCS-model [26]. Together with the fact which was already known to von Neumann that for type II and type III algebras automorphisms need not be implemented by an inner unitary, new attempts to quantum ergodic theory became possible.

Let us first concentrate on spectral theory. Let us assume that the automorphism is inner. If there are invariant states at all they correspond to projection operators. However, for type II and type III algebras minimal projections do not exist. Therefore necessarily inner automorphism allow either many or no invariant states.

If the automorphism is not inner, then two questions appear. Is the automorphism unitarily implementable, and if so what is the equivalence class of these unitaries


and the stability of the spectrum inside this equivalence class. The first answer uses modular theory. For type 111 algebras the unitary can be constructed in analogy to Koopman's formalism based on the vector that implements the tracial state. This vector is unique up to unitaries belonging to the commutant (or equivalently to the algebra) (compare [13, p. 156]). So constructed unitary commutes with the modular conjugation J . In [13, Theorem VI, p. 151] von Neumann and Murray offer an antiisomorphism between the algebra and its commutant, and one can easily read off the antiunitary modular conjugation that implements this antiisomorphism. The Koopman construction fails if one cannot start with an invariant case as it may happen in the type Iloo and the type III case. As late as 1975 the generalization was proved by Haagerup [18, Theorem 3.2]: based on the modular conjugation J (extended to the type III case from the antiisomorphism in [13]) he showed that every automorphism can be implemented by a unitary such that J U J = U . Since the modular conjugation is unique up to unitaries belonging to the commutant, the spectral properties of the operator U are unique. In the case of Araki-Woods the unitary implementing time translation has, apart from the discrete eigenvalue corresponding to the GNS vector, an absolutely continuous infinitely degenerate spectrum. For the IIIz algebra considered by Powers the modular automorphism has a pure point spectrum, also infinitely degenerate and additive. In addition, we have space translations that allow a unique invariant state, but otherwise the corresponding unitary has again absolutely continuous spectrum.

However, the main subject of research in quantum ergodic theory was not the characterization of the spectrum. The central problem has no counterpart in classical ergodic theory. The decomposition of a state over an abelian algebra into pure states is unique, similarly the decomposition of an invariant state into pure invariant states. If our algebra is a type I factor with only inner automorphisms, invariant states only exist if the unitary has a point spectrum. If this point spectrum is not degenerate then the decomposition into extremal invariant states is unique. However, we have already realized that a good ergodic behaviour can only be expected for an outer automorphism. Again we can examine the spectrum of the corresponding unitary. But this .was not the center of research. The von Neumann algebra appeared only as a result of the representation of a C* algebra. Ergodicity was not a property of the automorphism on the von Neumann algebra but should be expressed as property of the C* algebra. Good ergodic behaviour means that the pure equilibrium states are the result of a unique decomposition into states extremal invariant with respect to time evolution, and that in the corresponding representation the von Neumann algebra should be a factor. It was the aim to characterize the properties that the time evolution had to satisfy to guarantee this fact.

Since space translations on the lattice are good examples which do not allow further decomposition of factor states, their property of strong asymptotic abelianess was examined and it was shown that it can be generalized to weak asymptotic abelianess and G-abelianess, without destroying the fact that the decomposition into extremal invariant states is coarser than the factor decomposition and therefore similarly to the factor decomposition is unique. Examples were constructed showing step

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by step how the theory was generalized [27]. The real problem in physics, what kind of asymptotic abelianess a given time evolution, constructed from a Hamiltonian with finite range interaction, really satisfies, remains unsolved except for a few exam- pies (the XY-model [28, 29], Galilei invariant theories at infinite temperature [30]).

3. Progress in classical ergodic theory and its counterpart in quantum theory

In classical theory the ergodic hypothesis can be sharpened by demanding that time correlation functions have to tend to 0 [8, p. 279]. This is e.g. guaranteed, if the spectrum of the unitary, which implements the time evolution, is apart from its unique eigenvector absolutely continuous. This is discussed in [8]. But, in addition, we can ask for the rate of convergence or for some kind of uniformity in the operators.

Let us concentrate on the second problem. The uniformity is guaranteed for the Bernoulli shift, its version as the baker transformation and quite generally for K (Kolmogorov)-systems [32]. Here we can find a past algebra that shrinks for negative times to 3.1 but increases for positive times to the total algebra.

DEFINITION 1. A K-system consists of (A, A0, r, o~) such that o~ e ~: = w, Un>0 rnAo = C, Nn<0 rn-40 : A.

The algebra-40 is calledpast algebra. As a consequence we have uniformity in clustering in the following sense.

LEMMA 1. For every A c .4 and n, 3E such that I w ( A r n B ) - w(A)og(B)l < ~IIAIIIIBII YB e Ao.

The guess that all K-systems are equivalent to Bernoulli shifts was disproven by Ornstein [33]. They permit rotations in the lattice points that can destroy the equivalence. Nevertheless Bernoulli shifts can serve as symbolic dynamics for all K-systems. The spectra of all unitaries implementing K-dynamics are the same: one pure point and otherwise infinite degenerate Lebesgue measure on the unit circle. Therefore no further information on the dynamics can be extracted from the spectral properties. As a finer characterization Kolmogorov introduced dynamical entropy [31, 32].

A quantum counterpart for the Bernoulli shift is provided by the lattice systems: the algebra at every lattice point is not abelian anymore but is replaced by a full matrix algebra. Also a K-system can be defined in the noncommutative context. In our Definition 1 there is no need to assume that .4, .Ao are abelian. To keep the close relation between the measure theoretic aspect and the algebra we demand that the invariant state is faithful. But nevertheless we are confronted with totally new features. Different from the classical situation no strategy exists, or at least it has not been found yet, to decide in a constructive way, whether a past algebra exists. In classical theory we can choose any finite partition B and construct its past Un<0 rnB. If we do not get the total algebra then we know that the system at least contains a K-system and it is a K-system, if the tail of this past algebra is trivial.


In quantum theory we may have a K-system, but for a badly chosen subalgebra the past algebra can be identical to the total algebra, because noncommutativity can make the algebra increase in an uncontrolled way [34].

The other new feature is the following: For the classical Bernoulli shift the imbedding of the past algebra into the total algebra is a tensor product, and that remains essentially true also for the generalizations. In the noncommutative case the theory of imbedding has opened a new area of research. Already in [12, p. 64] von Neumann characterizes the possibility to imbed a finite-dimensional matrix algebra Mk into another finite-dimensional matrix algebra MN. This is only possible if N = kl and the imbedding is a tensor algebra of the smaller algebra and its relative commutant. One of the important observations of von Neumann and Murray was the fact that if a von Neumann algebra is considered as subalgebra of a type Io~ algebra then again the I-factor is generated by the von Neumann algebra and its relative commutant (though due to topological problems not as tensor product) [12, Lemma 3.1.2]. This does not hold any more if the von Neumann algebra is imbedded into another von Neumann algebra. The relative commutant already defined in [12, Definition 11.1.1] can even become trivial. When von Neu- mann introduces this notion he does not give any example but remarks that "our information is incomplete" [12, p. 75]. Now our knowledge though still incomplete has increased substantially. For instance we know that the relative commutant can be trivial and new number theoretical restrictions appear [35]. It is also possible to construct K-systems with trivial relative commutant. Such an example is offered by the Price-Powers shift [36]. Therefore the variety of quantum K-systems is much larger than the freedom offered by Ornstein in the classical case.

It should be noted that the triviality of the relative commutant has also drastic consequences in improving the ergodic behaviour on the level of the underlying C'algebra. It can be shown that for almost all Price-Powers shifts with trivial relative commutant the tracial state is the only state invariant under translations [37] whereas this is not the case when the relative commutant is nontrivial.

There is another possibility to find connections with the classical Bernoulli shift and the dynamics on a noncommutative von Neumann algebra. When von Neumann justified the definition of entropy for mixed states over a quantum system [3] he did this by comparing with the entropy of a classical system, representing the measurement. The relevance of the classical system had to be controlled via a correction term, physically motivated by the irreversibility of the measurement. Looking for a counterpart of Kolmogorov's definition of a dynamical entropy it can be based on the repetition of measurements according to the dynamics. This repetition can be interpreted as a sequence of noncommutative algebras as it is done in [38]. It can also be related to a classical increasing algebra whose relevance has again to be controlled by a counter term. This is the strategy in [39] and [40]. In [41] it was observed that the abelian model even inherits the dynamics so that we compare a classical dynamical system with a quantum dynamical system.

If we return to the problems of the early days of quantum ergodic theory where the asymptotic abelianess was the main concern we find the following relations:

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In quantum K-systems with a faithful invariant state co the dynamics is weakly asymptotically abelian (it is even K-clustering in the sense of Lemma 1.)

If every subsystem allows a nontrivial abelian model together with some additional assumptions that are trivially satisfied for abelian systems, then the system is strongly asymptotically abelian [42].

4. Anosov systems

In classical theory a famous example is given by the Anosov system. Here the dynamics is continuous. A prototype is the geodesic flow on a compact connected hyperbolic Riemannian manifold together with two transversal flows. Anosov flow and transversal flows expressed as automorphisms, satisfy the Anosov relation Trot x = Ote-Xtx'Ct, Tt~y -~-OteXty~ t . The manifold admits two foliations that intersect transversally and allow to construct a past and a future algebra [32], respectively. This foliation permits again uniform clustering with respect to the past algebra, but even without referring to the past algebra we have for almost all operators

co(ArtB) - co(A)co(B) lim sup e_Zt < o~. (1)

t--+oo

The Lyapunov exponent X coincides with the dynamical entropy. The natural translation to noncommutative algebras seemed to be the definition

in [43]:

DEFINITION 2. Let A be a von Neumann algebra, co a faithful normal state on A, rt the dynamics, and Olj(S) additional automorphisms. If "~tOlj(S)T--t = Olj(e-Ljts) then the system admits an integrable Anosov structure with Lyapunov exponents ~,1 <-- ~,2 <-- . . . <_ ~,k,

The automorphisms are unitarily implementable. The joined spectrum of V implementing zt and Uj implementing 0q is apart from degeneracy {0, 0} (~){R, R+}, where Uj(s) f (x) = eisx f (x) and Vt f (x) = c t f (eXjtx). The commutation relations of the automorphisms imply on the level of the operators of the algebra,

XJ = 1 ln lim I loq(s)rtA- trAil (2) t s--,0 I lct j(s)A- All

In [44] a corresponding relation is expressed for the Weyl algebra: it is assumed that the dynamics takes place in some appropriate dense subalgebra of B(L2(R)), preferably the Weyl algebra itself. With L~ = otx~ + ~p~b,

1 ~-~ (L~, A) = lim sup t In II[Z~, A(t)]ll (3)

we define ~ --- sup ~.~. Therefore the estimate is now based on the derivative and not on the automorphism. Due to the flexibility in ~ it covers a larger class of examples.


Similarly as in the classical theory where the expanding foliation dominates the behaviour again it can be shown by an explicit example that the Lyapunov exponent is independent of A (of course in the permitted class of operators, so that every step is well defined).

The K-property of classical Anosov systems survives only as a vectorial K-fil- tering. We can assign to the spectral projections of Uj a corresponding subspace that is increasing or decreasing in time. With a smearing process this subspace of the Hilbert space corresponds to a subspace in the algebra. Whereas for an abelian algebra this subspace is also an subalgebra this step fails in the noncommutative case. Nevertheless it serves to formulate clustering in an appropriate variation of (1) [43, Theorem 3.6].

If we look for possible realizations of Anosov systems we observe:

1. The automorphisms rt and otj(s) can be inner automorphisms. If they are inner, then the corresponding subspaces are no subalgebras.

Proof: The isometries between projections corresponding to Uj belong to the appropriate subspace, but their adjoints belong to another one.

2. If the algebra is a type II1 algebra, then o/j cannot be inner.

Proof: The tracial state is invariant under ft. However, zt scales the projections corresponding to otj. This leads to a contradiction.

3. In the 111 situation the subspaces cannot correspond to subalgebras.

Proof: Assume that .40 C .4 is the past algebra. Then Ut~0 rt-40 = .4. Cor- respondingly -40 defines a projection P0 in .4' with st l i m t ~ = 1, therefore also lim/__,~ Tr(r/P0) = 1. On the other hand Tr(rtP0) = 0 is a continuous function with limt--,-~Tr(rtPo) = 0. Now [12, Theorem X] has defined an imbedding parameter corresponding to the expectation value of P, and [35] has shown that there exists a minimal value for this parameter. Again we have a contradiction.

4. If .4 is a type III algebra and rt is the modular automorphism group then aj with nontrivial )~j cannot exist.

Proof: From the spectral representation we can conclude that the state is also invariant under otj(s). However, every automorphism for which ~o is invariant has to commute with ft.

All these results seem to indicate that the concept of Anosov systems does not really work in the framework of noncommutative algebras. But the opposite is true, as Borchers [45] and Wiesbrock [46] have shown. We have only to replace the automorphism a by an endomorphism. Then we get automatically an Anosov structure if our modular automorphism is a K-system.

THEOREM 1 ([45, 46]). Let (.4, 09, zt, .40) be a modular K-system, i.e. rt is the modular automorphism on .4 with respect to 09 and rt-4o C -40 Yt < O. Then

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a) The GNS vector is cyclic and separating also for ,4o. b) Let U ° = e itt°t implement the modular automorphism for .4o and Ut = e iHt

the one for .4, then

[H, H0] = i ( H - Ho) = iG <_ 0 (4)

e iGs implements an endomorphism for s > O.

REMARK. The proof is based on the analyticity properties of the modular operator. It was inspired from the example of the Rindler wedge in relativistic quantum field theory. Here the modular automorphism of the wedge corresponds to the boost [47]. If the wedge is shifted in light-like direction the boost maps the shifted wedge into itself. Therefore G has to be taken as the generator of light-like shifts. Com- mutation relations between light-like shifts and boost satisfy the Anosov relation.

From the mathematical point of view we have the following inputs: Different from type II factors for type III factors vectors that are cyclic and separating for imbedded algebras are generic [48]. A variation of the Reeh-Schlieder theorem shows that the GNS vector belongs to this class. The modular operators can be considered as Friedrich extensions of the same quadratic form but with increasing domain. (Recall the results of von Neumann on the importance of the domain [5].) This reflects that H is increasing monotonically to H0. The exact result asks for the delicate connection between the modular operator with its analyticity properties and the modular conjugation.

In classical theory the K-property in the forward direction, i.e. the existence of an increasing past algebra, implements the K-property in the backward direction, i.e. the existence of a decreasing future algebra. In the language of Anosov flows we have two transversal foliations. The geometrical background of the Anosov structure in quantum field theory guarantees that again we have an Anosov structure for the Rindler wedge in both time (i.e. boost) directions. Whether this holds in general is an open problem. If the relative commutant of the past algebra is large enough it serves as the basis of a future algebra [29]. On the other hand, similar as for the Price-Powers shift, an example of a quantum system with Anosov structure can be constructed, where we have a trivial relative commutant. Nevertheless, this example allows again an Anosov structure in both time directions [29].

The example is based on q-deformed second quantization [49] of the evolution on the one-particle space. On this one-particle level we recover the difference between Hermitian and selfadjoint operators [5]. The modular automorphism acts as

"rta(f) = a(eipt f ) , y s a ( f ) = a (Us f ) , f E L2(~) (5)

with Us = e iGs, G = pe -x + e-~p. G is Hermitian but it has defect indices (1, 0) with eigenvector e - e X - ~ . Nevertheless, Us is well defined though not by the spectrum but by a differential equation. It satisfies e i H t U s e - i H t = Ue-es . After a variable transformation y = e ~ E R + we have

r t a ( f ( y ) ) = a(c t f (eey) ) , ~4a( f (y ) ) = a ( f ( y + s)), s > 0, (6)

which obviously defines an endomorphism.


Since we can define an endomorphism also on the commutant we obtain two operators on the one-particle level with defect indices (1, 0) and (0, 1). This makes it possible to melt the two to a self-adjoint operator, therefore on the total algebra .4 U.4 ' the endomorphism becomes an isomorphism that is generated by a unitary. We realize that the fact, that Hermitian operators need not be self-adjoint, which von Neumann wanted to get rid off [12, p. 11] opens new possibilities in quantum ergodic theory.

5. The crossed product in quantum ergodic theory Von Neumann's examples for type II and type III algebras are all crossed prod-

ucts of an abelian algebra with an ergodic automorphism group. In these constructions the basic importance of the modular automorphism group does not appear. However, we will observe that with some minor and nevertheless powerful modifi- cations we can recover much of the Anosov structure in these examples.

In [12] von Neumann and Murray offer an example of a type 111 algebra. Later it turned out that this example is isomorphic to all other hyperfinite II1 factors [16, Theorem XIV]. They start with a measure space [S,/z] where the measure can also be infinite. On this measure space they take an ergodic action of an at most countable infinite group ~ that allows a measure dr. They construct a v o n Neumann algebra R(/z, ~) (now one uses the notation .4 ~ G) over the Hilbert

space L2(d/z)® L2(dv) built by the operators

.f(x)~(x, g) = f(xg)~p(x, g), VhaP(x, g) = ~(x, gh). (7)

A general operator in the resulting von Neumann algebra can be written as j~(x, g) with f (x , g)~ L~(d/Z) for fixed g, satisfying the multiplication rule

?l?2 X, g) = E fl(x, gl)f2(xgl, g2). (8) g2gl=g

With the weight f (x , g) ~ f d/zf(x, 1) they obtain a trace over the algebra. The ergodicity of the action guarantees that R(/z, G) is a factor. Therefore the values that the trace take can decide whether we are dealing with a factor type I, II1, II~. The construction is very close to the Weyl algebra. We can take for S, R, d/z the standard Lebesgue measure and ~ = R with the shift as ergodic action. What fails is the fact that ~ is countable. Therefore the natural measure on G is again the Lebesgue measure. But as a consequence the projections in L~(d/Z) that are imbedded in R(/z, G) as the distribution E^(x)8(g) do not belong to the set on which the trace is defined. If there were no other projections, the algebra could be type III, but von Neumann had proven [6] that there are even minimal projections and the algebra is /3(LZ(R, d/z). Therefore von Neumann reduced the group ~ to the rationals Q and obtained a II~ algebra.

A famous example in this context is the rotation algebra where S is the unit circle and the automorphism group is Z acting as the translation with a fixed step

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size. If this size is irrational then the action is ergodic and we get a type II1 algebra. But it is more convenient to write La([0, 1], dx) ® 12 as L2([0, 1] @ [0, 1], d x d y ) so that the algebra is built by U = e 2zrix , V = e 2Jri(y+~px) . Then we can observe a strong similarity to the two-dimensional torus, which is a famous object in classical ergodic theory [32]. The von Neumann algebra inherits the classical actions as automorphisms, especially

O t T U k V , = U m V n, m = a k + b l , n = c k + d l , a , b , c , d ~ Z, a d - b c = l. (9)

In addition we have the actions

( ~ x ( f l ) u k v 1 = e i k ~ u k v l, ( T y ( t ~ ) u k v l = e i l f l u k v l . (10)

As for the classical algebra (or = 0) the eigenvectors of the matrix T define foliations, respectively directions, of the automorphisms or. They stay in Anosov relation to the automorphism otr, though the Anosov map cannot be expanded to a continuous flow. We should remark that the Lyapunov exponents which coincide with the eigenvalues of the matrix T define the Kolmogorov dynamical entropy in the classical situation. In the quantum situation they correspond to the dynamical entropy only in the sense of [38], whereas in the sense of [39] the dynamical entropy vanishes for almost all rotation parameters ot [37].

We turn to the type III example constructed by von Neumann [15]. We concentrate on the first one because it is in relation to an Anosov structure. The example is the following.

S is again the set of all real numbers, /z the common Lebesgue measure for Borel sets. The automorphism group is Q + ® Q acting as

Oep,~(x) = px + or. (11)

The group ~ contains the subgroup C0 = 1 ® Q and this group acts ergodically. Therefore R(/z, Q) is a type IIo~ algebra. The map or(p, 0) is not measure preserving and therefore according to [15, Theorems VIII, IX] the algebra is type III.

We can relate this example to further developments in the theory of crossed products. The first step was the generalization to crossed products of noncommutative algebras with an automorphism group. This procedure appears first in [50]. Here the aim is to construct an extension of the algebra in which the automorphism becomes inner. The main theorem in this context is as follows.

THEOREM 2. I f ol t is an asymptotically abelian automorphism over the algebra .4 and o9 an extremal invariant state over .4 then the natural extension o f oo over .4 ~a~g G is pure.

In [51] the crossed product construction is the main tool to characterize different types and their relation. Now it is not required that the groups being countable. The weight that generalizes the trace is defined as distribution. The main result is the following.


THEOREM 3. Let .4 be a type Iloo algebra. Then there exists an automorphism group ott such that the trace is unique only up to a scaling Tr o ol t = e t Tr. Con- structing the crossed product `4 ~%, R one obtains a factor III.

If the automorphism group is countable then .4 t>%, G is an algebra of type III0.

Let .4 be a type III factor and ott some modular automorphism. Then .4 r,% t R is a type Iloo algebra, not necessarily a factor. It is independent up to an isomorphism of the chosen modular automorphism group.

Repeating the constructions with the dual automorphisms leads as to algebras that are isomorphic to the starting ones.

If we now compare this result with von Neumann's example we recognize that we have exactly the situation of the type III0. or(O, 0) can be considered as automorphism on `40 t,'%O,o) Q mapping f ( x ) into f ( p x ) and Vo into Vpo. Further, or(p, 0) and or(l, a) satisfy the Anosov commutation relations. The tracial state over ..4o ~,~(1,,~) Q,

Tr(A~(x) Vo) -- 8~,o I f ( x ) d x (12) , /

is scaled to

/ if Tr oot(a,o)(f(x)Vo) = 8~,o f ( p x ) d x = 8~,o- f ( p x ) d x . P

(13)

Therefore the result in [51] also gives the type III. Notice, however, that the re- striction to the rationals is essential. For p, tr ~ R the crossed product construction works in the same way. However, the tracial weight is not defined. If we replace the Kronecker delta by the delta-function as it is done in [51] then we always get infinity. Since we can find a trace in a less obvious way in fact oe(p, 0) becomes an inner automorphism and our final algebra remains I~ with a nontrivial center.

We have learned from the observations of [45, 46], that we can have nontrivial results if we replace the automorphism by an endomorphism. We will do this now also in the crossed product construction.

Let .4 be an Anosov system equipped with an endomorphism os.4 C .A and an automorphism rt such that rto'sr_t = ae,~. We copy von Neumann's construction to define `4 tmo s R + as a von Neumann algebra on 7-t ® L 2 ( R + , d s ) . With the conditional expectation value as* defined via Tr(Aa~B) = Tr(a~*AB) YB the algebra is built by the operators

. 4~ ( s ) = (a*A~2)(s), V t ~ ( s ) = ®(s)g2(s + t), VTw(s) = ®(s - t ) ~ ( s - t). (14)

Therefore ad Vt implements the endomorphism and ad Vt* the conditional expecta-

tion. A tracial state has to satisfy Tr(VtVt*)= Tr(Vt*Vt ), together with Vt*Vt = ~)t as nontrivial projection the trace has to be c~. With the notation Vt* = V-t we can take

106 H. NARNHOFER

) Tr dtdt'g(t)g(f)Vt,4(t, f)Vt, OO

= f dtdt'g(t)g(f)8(t + t')Tr(a~*+tA(t, t'))ds®(s + t)®(s). (15)

This definition satisfies the necessary commutativity, it is defined on a dense set of operators, but in addition we observe that it is not invariant under the natural extension of the automorphism at. This implies that the automorphism is not inner and therefore the algebra must be type of IIo~. In the special example where ,,4 = L~(R+,dx), it is also possible to transfer the K-structure of ,,4 into the crossed product ,,4 ~os R+, and therefore provides us with an example of a K-system connected with an Anosov flow for the hyperfinite type Iloo factor [52].

Acknowledgement

I want to thank J. E. Woods for critical remarks and insider information.

[6] (34) [7] (39) [8] (41) [9] (43)

[10] (44) [11] (40) [12] (60) [13] (68) [14] (37) [15] (75)

REFERENCES

[1] John von Neumann: Collected works, A.H. Taub (ed.), Pergamon Press, Oxford, (1961), in this edition the following articles:

[2] (6) With D. Hilbert and L. Nordheim: Ueber die Grundlagen der Quantenmechanik, Math. Ann. 98 (1927), 1-30.

[3] (10) Thermodynamik quantenmechanischer Gesamtheiten, Goett. Nach. (1927), 273-291. [4] (22) Ueber die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen,

Math. Zschr. 30 (1929), 3-42. [5] (28) Allgemeine Eigenwerttheorie Herrnitescher Funktionaloperatoren, Math. Ann. 102 (1929), 370-427.

Die Eindeutigkeit der Schroedingerschen Operatoren, Math. Ann. 104 (1931), 570-578. Proof of the quasi-ergodic hypothesis, N.A.S. Proc. 18 (1932), 70-82. With B.O. Koopman: Dynamical systems of continuous spectra, N.A.S. Proc. 18 (1932), 255-263. Einige Saetze ueber messbare Abbildungen, Ann. Math. 33 (1932), 574-586. Zur Operatorenmethode in der klassischen Mechanik, Ann. Math. 33 (1932), 587-642. Physical applications of the ergodic hypothesis, N.A.S. Proc. 18 (1932), 263-266. With F. J. Murray: On rings of operators, Ann. Math. 37 (1936), 116-229. With F. J. Murray: On rings of operators II, Amer. Math. Soc. Trans. 41 (1937), 208-248. On infinite direct products, Compos. Math. 6 (1938), 1-77. On rings of operators III, Ann. Math. 41 (1940), 94-161.

[16] (89) With F. J. Murray: On rings of operators IV, Ann. Math. 44 (1943), 716-808. [17] A. Connes: Une classification des facteurs de type III, Ann. Ec. Norm. Sup. 6 (1975), 133-252; Classi-

fication of injective factors, Ann. Math. 104 (1976), 73-115; 86 (1967), 138-171. [18] U. Haagerup: The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271-283. [19] H. Araki, R. Haag and B. Schroer: Postulates of quantum field theory, Rev. Math. 25 (1963), 1834. [20] H Araki, E. J. Woods: Representations of the canonical commutation relations describing non-relativistic

free Bose gas, J. Math. Phys. 4 (1963), 637-662; A classification of factors, Publ. RIMS, Kyoto Univ. 2 (1968), 51-130.

[21] R. Haag: Local Quantum Physics, Springer, Berlin 1992. [22] J. Glimm: Type I C* algebras, Ann. Math. 73 (1961), 572-612. [23] H. Araki: Type of von Neumann algebras associated with free field, Prog. Theoret. Phys. 32 (1964),

844-854. [24] W. Driessler: On the type of local algebras in quantum field theory Commun. Math. Phys. 53 (1977),

295-298.


[25] R. T. Powers: Representations of uniformly hyperfinite algebras and their associated yon Neumann algebras, Ann. Math. 86 (1967), 138-171.

[26] W. Thirring: On the Mathematical Structure of the B.C.S.-Model, Commun. Math. Phys. 7 (1968), 181-189.

[27] D. Ruelle: Statistical Mechanics, Benjamin, New York 1969. [28] H. Araki, T. Matsui: Ground States of the XY-Model, Commun. Math Phys. 101 (1985), 213-245. [29] H. Narnhofer: Control on weak asymptotic abelianess with the help of the crossed product construction,

in Quantum Probability, Banach Center 43 (1998), 331-339. [30] H. Narnhofer and W. Thirring: Galilei Invariant Quantum Field Theory with Pair Interaction, J. Mod.

Phys. A 6 (1991), 2937-2970. [31] A. N. Kolmogorov: A new metric invariant of transitiv systems and automorphisms of Lebesgue spaces,

Dokl. Akad. Nauk 119 (1958), 861-864. [32] V. I. Arnold, A. Avez: Problemes ergodiques de la mecanique classique, Gauthier-Villars, Paris 1967. [33] D. S. Ornstein: Ergodic theory, randomness and dynamical systems, Yale Math. Monographs 5, Yale

Univ. 1974. [34] H. Narnhofer, A. Pflug and W. Thirring: Mixing and entropy increase in quantum systems, Scuola Normale

Sup. Pisa (1989), 597-626. [35] V. F. R. Jones: Index for subfactors, Invent. Math. 72 (1983), 1-25. [36] R. T. Powers: An index theorem for semigroups of *-endomorphisms of B(H) and type I l l factors,

Canad. J. Math. 40 (1988), 86-114, G. 1. Price: Shifts of I l l factors, ibid. 39 (1987), 492-511. [37] H. Narnhofer and W. Thirring: C* dynamical systems that are asymptotically highly anticommutative,

Lett. Math. Phys. 35 (1995), 145-154. [38] R. Alicki and M. Fannes: Defining quantum dynamical entropy, Lett. Math. Phys. 32 (1994), 75-82. [39] A. Connes and E. Stoermer: Entropy of automorphisms of II1 von Neumann algebras, Acta Math. 134

(1975), 289-306. [40] A. Connes, H. Narnhofer and W. Thirring: Dynamical Entropy for C* algebras and von Neumann algebras,

Commun. Math. Phys. 112 (1987), 691-719. [41] J. L. Sauvageot and J. P. Thouvenot: Une nouvelle definition de l'entropic dynamique des systemes

noncommutatifs, Commun. Math Phys. 145 (1992), 411-423. [42] F. Benatti and H. Narnhofer: Strong asymptotic abelianess for entropic K-systems, Commun. Math. Phys.

136 (1991), 231-250; Strong clustering in Type III entropic K-Systems, Mh. Math. 124 (1996), 287-307. [43] G. G. Emch, H. Narnhofer, G. L. Sewell and W. Thirring: Anosov actions on noncommutative algebras,

Jour. Math. Phys. 35/11 (1994), 5582-5599. [44] H. R. Jauslin, O.Sapin, S.Guerin and W. F. Wreszinski: Upper quantum Lyapunov exponent for parametric

oscillators, J. Math. Phys. 44 (2004), 4377-4385. [45] H. J. Borchers: On modular inclusion and spectral condition, Lett. Math. Phys. 27 (1993), 311-324; 184

(1997), 683-685. [46] H. J. Wiesbrock: Half sided modular inclusions of von Neumann algebras, Commun. Math. Phys. 157

(1993), 83-92; 184 (1997), 683-685. [47] J. Bisognano and E. H. Wichmann: On the duality of an herrnitian scalar field, Jour. Math. Phys. 16

(1975), 985-1007; On the duality condition for quantum fields, ibid. 17 (1976), 303-312. [48] J. Dixmier and O. Marechal: Vecteurs totalisateurs d'une algebre de von Neumann, Commun. Math. Phys.

22 (1971), 44-50. [49] D. Shlyakhtenko: Free quasifree states, Pac. J. Math. 177 (1997), 329-368. [50] S. Doplicher, D. Kastler and D. W. Robinson: Covariance algebras in field theory and statistical mechanics,

Commun. Math. Phys. 3 (1996), 1-28. [51] M. Takesaki: Duality for crossed products and the structure of yon Neumann algebras of type III, Acta

Math. 131 (1973), 249-310. [52] H. Narnhofer: in preparation.

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Roots for quantum ergodic theory invon Neumann's work

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