A COUNTEREXAMPLE IN VON NEUMANN ALGEBRA DYNAMICS TIM AUSTIN ABSTRACT. In this essay we construct a noncommutative von Neumann dynamical system with certain pathological properties from the view- point of analogies with classical ergodic theory. These relate to obstruc- tions to the existence of special kinds of subsystem or subextension. We will build our example as an extension of a commutative dynamical sys- tem via a unitary cocycle acting on a Hilbert bundle, and will see that the well-known possibility of pathological behaviour for such unitary cocy- cles can, with a little care, be translated into pathological behaviour for von Neumann dynamical system extensions. CONTENTS 1. Introduction 1 2. The main construction 2 2.1. Hilbert bundles, cocyles and tensor products 2 2.2. The construction of the example 3 3. Application: extending globally-fixed Abelian subalgebras 6 Acknowledgements 8 References 8 1. INTRODUCTION We study here a particular von Neumann dynamical system: an ordered triple (M; θ; ®) consisting of a von Neumann algebra M equipped with a faithful normal state θ (serving as the analog of the measure) and a θ- preserving automorphism ®. In particular, we construct an example of such a system, naturally represented on a particular Hilbert space such that the automorphism is given by some unitary on that space, which exhibits var- ious ‘strictly noncommutative’ behaviour, in that it violates the obvious noncommutative analogs of certain standard results of ergodic theory for probability-preserving systems. Our example is obtained as an extension 1 2 TIM AUSTIN of a classical (that is, commutative) measurable dynamical system via a carefully-chosen Hilbert bundle and unitary cocycle over that system; the ‘essentially noncommutative’ behaviour of this extension will follow from that of the cocycle. After specifying our example, we will consider the possibility that the de- mand of global invariance under an automorphism of a von Neumann al- gebra can obstruct the extensions of a subalgebra. Suppose we are given a von Neumann dynamical system (M; θ; ®) spatially represented on some Hilbert space H, and suppose further that M contains some ®-invariant Abelian subalgebra N. Is it possible that N is not maximal among all Abelian subalgebras of M, but is maximal among those that are ®-invariant? We will show that the answer is Yes; in doing so, we will obtain a bound on the applicability of a certain model of Hilbert space operator. 2. THE MAIN CONSTRUCTION 2.1. Hilbert bundles, cocyles and tensor products. Let (­; §; ¹; ¿) be a Lebesgue probability-preserving system and H0 some fixed complex Hilbert space (which we will later take finite-dimensional). Henceforth, we will let M be the algebra L1 (¹) of bounded measurable functions from ­ to B(H0) B(H0), defined up to ¹-almost everywhere equality. We will refer to these as operator cocycles. As is standard, M can be represented through its natural pointwise action on H := L2 (¹); given A 2 M we will sometimes H0 1 write MultA 2 B(H) for this operator. In this picture we can identify LC (¹) as the centre of M and write M as the direct integral in the obvious way: Z © M = B(H!) ¹(d!); ­ where we write H! for a copy of H0 associated with the point ! 2 ­. In addition, M also carries the invariant faithful normal trace θ given by Z θ(A) = Tr(A(!)) ¹(d!) ­ for A : ­ !B(H0) in M. The unitary members of M correspond precisely to unitary cocycles: mea- surable maps © : ­ !U(H0) to the unitary group of H0. Given such a © let us also define the unitary operator U ¿;© on H by ¿;© ¡ # ¢ U f(!) = ©(¿(!))f(¿(!)) = (Mult© ± ¿ )f (!); writing ¿ # for the unitary operator on H given by composition with ¿. We will refer to a unitary U ¿;© constructed in this way as a twisted composition VON NEUMANN DYNAMICS 3 operator over ¿. This definition can clearly be extended to any operator of # the form MultA ± ¿ for some operator cocycle A 2 M; we will refer to these as generalized weighted composition operators over ¿. Given a unitary U 2 U(H) for any Hilbert space H and a von Neumann algebra M on H with UMU ¤ = M, we will write ®U for the automorphism on M defined by conjugation by U; for the twisted composition unitaries U ¿;© constructed above we will abbreviate ®U ¿;© to ®¿;©. 2.2. The construction of the example. We now specialize to the particular dynamical systems and cocycles that we will use later. We take the fibre 2 space H0 to be C (although the arguments that follow could be adapted to any space of finite dimension greater than 1) and so U(H0) = U(2). For our underlying probability-preserving system (­; §; ¹; ¿) we take the 1 1 1 1 ( 4 ; 4 ; 4 ; 4 )-Bernoulli shift system given by the coordinate right-shift on f1; 2; 3; 4gZ with its usual product measure space structure. (We will see 1 1 below that there is a related construction over the simpler ( 2 ; 2 )-Bernoulli shift, but at a certain point this then necessitates a slightly longer argument.) th Write ¼i : ­ ! f1; 2; 3; 4g for the projection onto the i coordinate. Our unitary cocycles © will depend only on the zeroth coordinate in ­, and so take at most four distinct values. Moreover, these will be of the form V1, ¡1 ¡1 V2, V3 := V1 and V4 := V2 , so that ©(!) = V¼0(!). It is so that we can introduce two distinct unitaries and also their inverses that we work over a shift alphabet of size four. We will not specify particular V1 and V2 in U(2), but will show that a uni- form random pair is almost sure to have the properties we will call on later. This relies on a manipulation of the dynamics of these cocycles through the following lemma. Lemma 2.1. Let V1, V2 2 U(2). Consider the set Wn of all words of length N in the free group F2 on two generators, and, given such a word w, let ¡1 ¡1 wV1;V2 (w) be the corresponding word in V1, V2, V1 and V2 . Then for the usual Haar measure on U(2)£U(2), almost every pair (V1;V2) is such that the empirical measures 1 X ± wV1;V2 (w) jWnj w2Wn converge vaguely to the Haar measure on U(2) as n ! 1. Remark The proof we give below relies on a much more abstract result about ergodic actions of free groups: the Nevo-Stein ergodic theorem ([3]). However, it might be interesting to seek a more elementary approach to this 4 TIM AUSTIN special case, and such an elementary approach might then also remove the ¡1 ¡1 need to include the inverses V1 , V2 . C 2 Proof We can identify the words wV1;V2 (w) as the images of 1C under the action of F2 on U(2) for which the two generators are sent to left- multiplication by V1 and V2 respectively. It is now routine to verify (for example, by reducing the question to a consideration of the irreducible rep- resentations of U(2) through an appeal to the Peter-Weyl Theorem) that this action is ergodic for almost every choice of (V1;V2); and now the Nevo- Stein free group ergodic theorem (see [3]) gives pointwise convergence of the ergodic averages corresponding to the points wV1;V2 (w), w 2 Wn, and so also the desired vague convergence of the empirical measures. (Note that in the first instance the Nevo-Stein Theorem gives this convergence only for the sets Wn corresponding to even lengths n; however, it is clear that in our case there is enough rigidity in the action that this implies the full result.) ¤ Let us now write out explicitly the form of the images (®¿;©)n(A) of A for an operator-valued cocycle A : ­ !B(H0): (®¿;©)nA(!) = ©(¿(!))©(¿ 2(!)) ¢ ¢ ¢ ©(¿ n(!))A(¿ n(!))©(¿ n(!))¤ ¢ ¢ ¢ ©(¿ 2(!))¤©(¿(!))¤ n ¤ ¤ ¤ = V!1 V!2 ¢ ¢ ¢ V!n A(¿ (!))V!n ¢ ¢ ¢ V!2 V!1 : From this, we can deduce the particular property that we will need for our counterexamples. Lemma 2.2. Suppose that k ¸ 0 and A is the algebra of all clopen subsets of ­ depending only on the coordinates ¡k; ¡k + 1; : : : ; k, and that A : ­ ! Mat2£2(C) is a measurable operator-valued cocycle that is actually A-measurable. Suppose also that C ⊆ ­ is a finite dimensional cylinder of the form f! : ¼i(!) = ´i for ¡ m · i · mg. Then as n ! 1 the conditional distribution of (®¿;©)n(A) inside the set C converges vaguely to the combined distribution of a matrix sampled from the distribution of A conjugated by an independent uniformly distributed unitary in U(2). Proof This is a straightforward consequence of our above calculation. By enlarging either m or k as necessary, we may assume they are equal (for if the conclusion holds for each of a partition of C into cylinder sets depending on more coordinates, then it certainly holds for C itself). If n ¸ 2k +1 then (®¿;©)n(A) is given by ¤ ¤ ¤ V!1 V!2 ¢ ¢ ¢ V!n A(!n¡k;!n+1¡k;:::;!n+k)V!1 V!2 ¢ ¢ ¢ V!n : We may re-write this as ¤ ¤ ¤ Wn;1Wn;2Wn;3A(!jfn¡k;n¡k+1;:::;n+kg)Wn;3Wn;2Wn;1; VON NEUMANN DYNAMICS 5 where Wn;1 := V!1 V!2 ¢ ¢ ¢ V!k Wn;2 := V!k+1 V!k+2 ¢ ¢ ¢ V!n¡k¡1 ; Wn;3 := V!n¡k V!n¡k+1 ¢ ¢ ¢ V!n : Now, as n ! 1, by Lemma 2.1 the unitary Wn;2 converge vaguely in dis- tribution to the Haar measure on U(2) (for their distribution is clearly pre- cisely that of the images of words of length n ¡ 2k under the action of F2 considered in Lemma 2.1).