aa r X i v : . [ m a t h . OA ] J u l A NOTE ON THE CLASSIFICATION OF GAMMA FACTORS
ROM ´AN SASYK
Abstract.
One of the earliest invariants introduced in the study of finite vonNeumann algebras is the property Γ of Murray and von Neumann. In this notewe prove that it is not possible to classify separable II factors satisfying theproperty Γ up to isomorphism by a Borel measurable assignment of countablestructures as invariants. We also show that the same holds true for the full II factors. Introduction
In this note we continue with the line of research initiated by the author incollaboration with A. T¨ornquist in [19], [20] and [21], where we applied the notionof
Borel reducibility from descriptive set theory to study the complexity of theclassification problem of several different classes of separable von Neumann algebras.Recall that if E and F are equivalence relations on standard Borel spaces X and Y , respectively, we say that E is Borel reducible to F if there is a Borel function f : X → Y such that ( ∀ x, x ′ ∈ X ) xEx ′ ⇐⇒ f ( x ) F f ( x ′ ) , and if this is the case we write E ≤ B F . Thus if E ≤ B F then the points of X can be classified up to E equivalence by a Borel assignment of invariants that wemay think of as F -equivalence classes. E is smooth if it is Borel reducible to theequality relation on R . While smoothness is desirable, it is most often too muchto ask for. A more generous class of invariants which seems natural to considerare countable groups, graphs, fields, or other countable structures, considered upto isomorphism. Thus, following [14], we will say that an equivalence relation E is classifiable by countable structures if there is a countable language L such that E ≤ B ≃ Mod( L ) , where ≃ Mod( L ) denotes isomorphism in Mod( L ), the Polish spaceof countable models of L with universe N .In [20] it was proved that the isomorphism relation in the set of finite von Neu-mann algebras is not classifiable by countable structures. Nonetheless, it can cer-tainly be the case that some subclasses of finite factors are possible to classify bycountable structures. For instance, Connes’ celebrated Theorem [3], says that theset of infinite dimensional injective finite factors has only one element on its iso-morphism class, namely the hyperfinite II factor R . In contrast with the injectivecase, in this note we show that it is not possible to obtain a reasonable classificationup to isomorphisms for a well studied family of finite factors that includes R . Inorder to state our results we observe first that the set of finite factors can be split in Mathematics Subject Classification.
Key words and phrases. von Neumann algebras; descriptive set theory; Gamma factors.The author acknowledges support from the following grants: PICT 2012-1292 (ANPCyT),and UBACyT 2011-2014 (UBA). two disjoint subsets: those who satisfy the property Γ of Murray and von Neumannand those who are full . The first set contains the hyperfinite II factor R and moregenerally, the class of McDuff factors, i.e. those factors of the form M ⊗ R for M aII factor. On the other hand the set of full factors contains the free group factors L ( F n ). In this article we show that the II factors constructed in [20] are full. Asa consequence, Theorem 7 in [20] strengthens to prove: Theorem 1.1.
The isomorphism relation for full type II factors is not classifiableby countable structures. It remained then to analyze the complexity of the classification of II factorswith the property Γ. In this note we address this problem by showing that: Theorem 1.2.
The isomorphism relation for McDuff factors is not classifiable bycountable structures.
An immediate consequence is:
Corollary 1.3.
The isomorphism relation for type II factors satisfying the prop-erty Γ of Murray and von Neumann is not classifiable by countable structures. We end this introduction by mentioning that the study of the connections be-tween logic and operator algebras has recently attracted many researchers fromboth fields. As a consequence, in the past five years there has been a burst ofactivity in proving results along the lines of the ones presented in this note andfirst unveiled in [19], [20] and [21]. We refer the reader who wants to learn more onthese exciting new developments to the recent survey of I. Farah [9].2.
Gamma factors
We start by recalling the definitions of the objects we study in this article. Let H be an infinite dimensional separable complex Hilbert space and denote by B ( H ) thespace of bounded operators on H , which we give the weak topology. A separablevon Neumann algebra is a weakly closed self-adjoint subalgebra of B ( H ). The setof von Neumann algebras acting on H is denoted vN( H ). A von Neumann algebra M is said to be finite if it admits a finite faithful normal tracial state, i.e. a linearfunctional τ : M → C such that: τ ( x ∗ x ) ≥ τ ( x ∗ x ) = 0 iff x = 0, τ (1) = 1 , τ ( xy ) = τ ( yx ) and the unit ball of M is complete with respect to the norm givenby the trace k x k = τ ( x ∗ x ). If a finite von Neumann algebra is also a factor , i.e.its center is trivial, then it has a unique such a trace. A finite von Neumann factorthat is not a matrix algebra is called a type II factor. This terminology is dueto the general classification of von Neumann algebras according to types (see [4,Chapter 5.1] for an historical account of the theory of types).In this note we will be interested in II factors arising from the so called group-measure space construction , that we proceed to describe. For that, let G be acountably infinite discrete group which acts in a measure preserving way on a Borelprobability space ( X, µ ). For each g ∈ G and ζ ∈ L ( X, µ ) the formula σ g ( ζ )( x ) = ζ ( g − · x )defines a unitary operator on L ( X, µ ).We identify the Hilbert space H = L ( G, L ( X, µ )) with the Hilbert spaceof formal sums P g ∈ G ζ g ξ g , where the coefficients ζ g are in L ( X, µ ) and satisfy
LASSIFICATION OF GAMMA FACTORS 3 P g k ζ g k L ( X,µ ) < ∞ , and ξ g are indeterminates indexed by the elements of G . Theinner product on H is given by h X g ∈ G ζ g ( x ) ξ g , X g ∈ G ζ ′ g ( x ) ξ g i = X g ∈ G h ζ g , ζ ′ g i L ( X,µ ) . Both L ∞ ( X, µ ) and G act by left multiplication on H by the formulas f ( ζ g ( x ) ξ g ) = (( f ( x ) ζ g ( x )) ξ g ,u h ( ζ g ( x ) ξ g ) = σ h ( ζ g )( x ) ξ hg , where f ∈ L ∞ ( X, µ ), ζ g ( x ) ∈ L ( X, µ ) and g, h ∈ G . Thus if we denote by FS theset of finite sums, FS = { X g ∈ G f g u g : f g ∈ L ∞ ( X, µ ) , f g = 0 , except for finitely many g } , then each element in FS defines a bounded operator on H . Moreover, multiplicationand involution in FS satisfy the formulas( f g u g )( f h u h ) = f g σ g ( f h ) u gh and ( f u g ) ∗ = σ g − ( f ∗ ) u g − and so FS is a ∗ -algebra. By definition, the group-measure space von Neumannalgebra is the weak operator closure of FS on B ( H ) and it is denoted by L ∞ ( X, µ ) ⋊ σ G . The trace on FS , defined by τ ( X g ∈ G f g u g ) = Z X f e dµ, extends to a faithful normal tracial state in L ∞ ( X ) ⋊ σ G by the formula τ ( T ) = h T ( ξ e ) , ξ e i , where e represents the identity of G . Definition 2.1 (Murray-von Neumann [15]) . A finite von Neumann algebra M hasthe property Γ if given x , . . . , x n ∈ M, and ε > u ∈ U ( M ) , τ ( u ) = 0such that k x i u − ux i k < ε, for all 1 ≤ i ≤ n. It follows immediately from its definition that the hyperfinite II factor R is aΓ-factor. Moreover, it is clear that any finite factor of the form M ⊗ N with N aΓ-factor is also a Γ-factor. In particular, McDuff factors , i.e. factors of the form M ⊗ R , are Γ-factors. The paradoxical decomposition of the free groups F n , n ≥ L ( F n ), n ≥ G is a discrete ICC group and L ( G ) has the propertyΓ, then G is inner amenable, (so in particular, free groups are not inner amenable).That the converse of Effros’ theorem is false is a recent result of Vaes [24].If M is finite von Neumann algebra with trace τ , Aut( M, τ ), the set of τ -preserving automorphisms of M is a Polish group. A basis for that topology isgiven by the sets V T,a ,...,a n ,ε = { S ∈ Aut(
M, τ ) : || S ( a i ) − T ( a i ) || ≤ ε, ∀ ≤ i ≤ ICC stands for infinite conjugacy classes. G is ICC if and only if L ( G ) is a factor. ROM´AN SASYK n, a i ∈ M } . Inn( M ) denotes the set of inner automorphisms of M , i.e. those ofthe form Ad ( u ), u ∈ U ( M ). Definition 2.2 (Connes [2]) . A finite von Neumann algebra M is full if Inn( M ) isclosed in Aut( M ).In [2, Corollary 3.8], Connes showed that a II factor M is full if and only if M does not have the property Γ. It follows that for each n ≥ L ( F n ) is a full factor.In order to discern when group measure space von Neumann algebras are full weneed the following: Definition 2.3 (Schmidt [22]) . Let G be a discrete group and let σ be an ergodicmeasure preserving action of G on a probability space ( X, µ ). A sequence ( B n ) n ∈ N of measurable subsets of X is asymptotically invariant if µ ( B n △ σ g ( B n )) → , for all g ∈ G. The sequence is trivial if µ ( B n )(1 − µ ( B n )) → . The action σ is strongly ergodic if every asymptotically invariant sequence is trivial.The relation between strong ergodicity and fullness has been studied by severalauthors. For the purpose of this note, it is enough to mention the following Theoremof Choda [1]: Theorem 2.4.
Let G be a discrete group that is not inner amenable, and let σ bea strongly ergodic measure presearving action of G on a probability space ( X, µ ) .Then L ∞ ( X, µ ) ⋊ σ G is a full factor.Remark . The condition that is really used in the proof of Theorem 2.4 is that L ( G ) is full.It is known that a group is amenable if and only if it does not admit stronglyergodic actions [22], while a group has the property (T) of Kazdhan if and only ifevery m.p. ergodic action of it is strongly ergodic [6]. We describe now a concreteexample of a strongly ergodic action of F that we will use in this work. Since F can be identified with the finite index subgroup of SL (2 , Z ) generated by thematrices (cid:26)(cid:20) (cid:21) , (cid:20) (cid:21)(cid:27) (see [7, II.B.25]), it follows that F naturally acts on T . Lets denote such action by σ and by T a , T b the automorphisms correspondingto the generators a , b of F . This action is clearly measure preserving, and oneof the main results in [22] is that σ is strongly ergodic. Inspired by earlier workof Gaboriau and Popa and T¨ornquist ([12], [23]), in [20] we used this action asthe starting point for showing that II factors are not classifiable by countablestructures. More precisely, the setExt( σ ) = { S ∈ Aut( T , µ ) : T a , T b , S generates a free action of F } was shown in [23, §
3] to be a dense G δ subset of Aut( T , µ ). Thus Ext( σ ) is astandard Borel space. For each S ∈ Ext( σ ) denote by σ S the corresponding F action and M S ∈ vN( L ( F , L ( T , µ )) the corresponding group-measure space vonNeumann algebra M S = L ∞ ( T ) ⋊ σ S F . LASSIFICATION OF GAMMA FACTORS 5
In [20] the author and T¨ornquist showed:
Theorem 2.6.
The equivalence relation on
Ext( σ ) given by S ≃ F II1 S ′ if M S isisomorphic to M S ′ is not classifiable by countable structures. In [20] it was shown that S → M S is a Borel map from Ext( σ ) to vN( L ( F , L ( T , µ ))).Thus Theorem 1.1 is an immediate consequence of the previous theorem and of thenext: Lemma 2.7.
For each S ∈ Ext ( σ ) , M S is a full factor.Proof. Let ( B n ) n ∈ N be an asymptotically invariant sequence for the F -action σ S .Then ( B n ) n ∈ N is an asymptotically invariant sequence for the action restrictedto the subgroup generated by { T a , T b } . By construction, this is the F -action σ described above, thus it is strongly ergodic by [22, § B n ) n ∈ N istrivial and then σ S is strongly ergodic. Since F is not inner amenable, the resultnow follows from Theorem 2.4. (cid:3) In order to prove Theorem 1.2 we require the following Theorem of Popa ([18,Theorem 5.1]):
Theorem 2.8. If M and M are full type II factors such that M ⊗ R , is iso-morphic to M ⊗ R then there exists t ∈ R > such that M is isomorphic to M t .Remark . By interchanging the roles of M and M one can assume that t ∈ (0 , M t is by definition the type II factor pM p where p ∈ P ( M )is any projection of trace equal to t in M . Theorem 2.10.
The assignment M S → M S ⊗ R is a Borel reduction of ≃ F II1 toisomorphism of McDuff factors.Proof.
It is fairly straightforward to prove that the map M S → M S ⊗ R is a Borelassignment (see for instance [13, Corollary 3.8]) . We are left to show that if M S ⊗ R is isomorphic to M S ′ ⊗ R , then M S is isomorphic to M S ′ .Let us fix S, S ′ ∈ Ext( σ ). Lemma 2.7 shows that M S and M S ′ are full factors.By Theorem 2.8, M S ⊗ R is isomorphic to M S ′ ⊗ R if and only if there exists t > M S ′ is isomorphic to ( M S ) t . The proof is over once we show that t = 1.For this we make use of the celebrated theorem of Popa on II factors with trivialfundamental group [17], [16], (see also Connes’s account in the Bourbaki S´eminaire[5]). Indeed by [16, Proposition], there exists a projection p ∈ P ( L ∞ ( T )), τ ( p ) = t ,such that the inclusion of von Neumann algebras ( L ∞ ( T ) ⊂ L ∞ ( T ) ⋊ σ S ′ F ) is iso-morphic to the inclusion of von Neumann algebras pL ∞ ( T ) ⊂ p (cid:0) L ∞ ( T ) ⋊ σ S F (cid:1) p .Feldman-Moore’s Theorem [10] applies to conclude that the action σ S is stable or-bit equivalent to the action σ S ′ , with compression constant c = t . Since F hasnon trivial Atiyah’s ℓ -betti numbers, Gaboriau’s Theorem on ℓ -betti numbers fororbit equivalence relations [11, Theorem 3.12] then implies that t = 1. (cid:3) Proof of Theorem 1.2.
Since M S → M S ⊗ R is a Borel reduction of ≃ F II1 to iso-morphism of McDuff factors and the equivalence relation ≃ F II1 is not classifiableby countable structures, it follows that the equivalence relation of isomorphism ofMcDuff factors is not classifiable by countable structures. (cid:3)
ROM´AN SASYK
References
1. M. Choda,
Inner amenability and fullness , Proceedings of the American Mathematical Society (1982), no. 4, 663–666.2. A. Connes, almost periodic states and factors of type III , Journal of Functional Analysis (1974), 415–445.3. , Classification of injective factors, cases II , II ∞ , III λ , λ = 1, Annals of Mathematics (1976), 73–115.4. , Noncommutative geometry , Academic Press, 1994.5. A. Connes,
Nombres de Betti L et facteurs de type II (d’apr`es D. Gaboriau et S. Popa) ,Ast´erisque (2004), no. 294, ix, 321–333.6. A. Connes and B. Weiss, Property T and asymptotically invariant sequences , Israel Journalof Mathematics (1980), 209–210.7. P. de la Harpe, Topics in geometric group theory , Chicago Lectures in Mathematics, Universityof Chicago Press, 2000.8. E. Effros,
Property Γ and inner amenability , Proceedings of the American MathematicalSociety (1975), 483–486.9. I. Farah, Logic and operator algebras , Proceedings of the International Congress ofMathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 15–39.10. J. Feldman and C. C. Moore,
Ergodic equivalence relations, cohomology, and von Neumannalgebras.I and II , Trans. Amer. Math. Soc. (1977), no. 2, 289–359.11. D. Gaboriau,
Invariants ℓ de relations d equivalence et de groupes , Publ. math., Inst. HautesEtud. Sci (2002), 93–150.12. D. Gaboriau and S. Popa, An uncountable family of nonorbit equivalent actions of F n , Journalof the American Mathematical Society (2005), 547–559.13. U. Haagerup and C. Winsløw, The Effros-Mar´echal topology in the space of von Neumannalgebras. I , American Journal of Mathematics (1998), no. 3, 567–617.14. G. Hjorth,
Classification and orbit equivalence relations , Mathematical Surveys and Mono-graphs, vol. 75, American Mathematical Society, 2000.15. F. Murray and J. von Neumann,
On rings of operators, IV , Annals of Mathematics (2) (1943), 716–808.16. S. Popa, On the fundamental group of type II factors , Proceedings of the National Academyof Sciences (2004), no. 3, 723–726.17. , On a class of type II factors with Betti numbers invariants , Annals of Mathematics (2006), 809–899.18. , On Ozawa’s property for free group factors , International Mathematics ResearchNotices. IMRN (2007), Art. ID rnm036, 10.19. R. Sasyk and A. T¨ornquist, Borel reducibility and classification of von Neumann algebras ,Bulletin of Symbolic Logic (2009), no. 2, 169–183.20. , The classification problem for von Neumann factors , Journal of Functional Analysis (2009), 2710–2724.21. ,
Turbulence and Araki-Woods factors , Journal of Functional Analysis (2010),2238–2252.22. K. Schmidt,
Asymptotically invariant sequences and an action of
SL(2 , Z ) on the -sphere ,Israel Journal of Mathematics (1980), 193–208.23. A. T¨ornquist, Orbit equivalence and actions of F n , Journal of Symbolic Logic (2006),265–282.24. S. Vaes, An inner amenable group whose von Neumann algebra does not have propertyGamma , Acta Mathematica (2012), 389–394.
Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universi-dad de Buenos Aires, ArgentinaandInstituto Argentino de Matem´aticas-CONICET, Saavedra 15, Piso 3 (1083), BuenosAires, Argentina
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