On cocycle superrigidity for Gaussian actions
aa r X i v : . [ m a t h . OA ] M a r ON COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS
JESSE PETERSON AND THOMAS SINCLAIR
Abstract.
We present a general setting to investigate U fin -cocycle superrigid-ity for Gaussian actions in terms of closable derivations on von Neumann al-gebras. In this setting we give new proofs to some U fin -cocycle superrigidityresults of S. Popa and we produce new examples of this phenomenon. Wealso use a result of K. R. Parthasarathy and K. Schmidt to give a necessarycohomological condition on a group representation in order for the resultingGaussian action to be U fin -cocycle superrigid. Introduction
A central motivating problem in the theory of measure-preserving actions ofcountable groups on probability spaces is to classify certain actions up to orbitequivalence, i.e. , isomorphism of the underlying probability spaces such that theorbits of one group are carried onto the orbits of the other. When the groups areamenable this problem was completely settled in the early ’80s (cf. [13, 14, 21, 9]):all free ergodic actions of countable, discrete, amenable groups are orbit equivalent.The nonamenable case, however, is much more complex and has recently seen aflourish of activity including a number of striking results. We direct the reader tothe survey articles [37, 47] for a summary of these recent developments.One breakthrough which we highlight here is Popa’s use of his deformation/rigiditytechniques in von Neumann algebras to produce rigidity results for orbit equivalence(cf. [31, 33, 34, 35, 36, 38, 39]). One of the seminal results using these techniquesis Popa’s Cocycle Superrigidity Theorem [36, 38] (see also [17] and [48] for more onthis) which obtains orbit equivalence superrigidity results by means of untwistingcocycles into a finite von Neumann algebra. In order to state this result we recalla few notions regarding groups.A subgroup Γ < Γ is w-normal if there is a chain of subgroups Γ = Γ < Γ < · · · < Γ β = Γ for some ordinal β such that ( S α ′ <α Γ α ′ ) ⊳ Γ α for all α ≤ β . Agroup Γ is w-rigid if there exists an infinite w-normal subgroup Γ such that thepair (Γ , Γ ) has relative property (T). If U is a class of Polish groups then a free,ergodic, measure-preserving action of a countable discrete group Γ on a standardprobability space ( X, µ ) is said to be U -cocycle superrigid if any cocycle for theaction Γ y ( X, µ ) which is valued in a group contained in the class U must becohomologous to a homomorphism. U fin is used to denote the class of Polish groupswhich arise as closed subgroups of the unitary groups of II factors. In particular,the class of compact Polish groups and the class of countable discrete groups areboth contained in U fin . The notions of w-normality, w-rigidity, and the class U fin The first author’s research is partially supported by NSF Grant 0901510 and a grant from theAlfred P. Sloan Foundation. are due to Popa (cf. [33, 36]).
Popa’s Cocycle Superrigidity Theorem ([36], [38]) (for Bernoulli shift actions).Let Γ be a group which is either w-rigid or contains a w-normal subgroup which isthe direct product of an infinite group and a nonamenable group, and let ( X , µ )be a standard probability space. Then the Bernoulli shift action Γ y Π g ∈ Γ ( X , µ )is U fin -cocycle superrigid.The proof of this theorem uses a combination of deformation/rigidity and inter-twining techniques that were initiated in [32]. Roughly, if we are given a cocycleinto a unitary group of a II factor, we may consider the “twisted” group algebrasitting inside of the group-measure space construction. The existence of rigiditycan then be contrasted against natural malleable deformations from the Bernoullishift in order to locate the “twisted” algebra inside of the group-measure spaceconstruction. Locating the “twisted” algebra allows us to “untwist” it, and, in sodoing, untwist the cocycle in the process.The existence of such s-malleable deformations (introduced by Popa in [34, 35])actually occurs in a broader setting than the (generalized) Bernoulli shifts withdiffuse core, but it was Furman [17] who first noticed that the even larger classof Gaussian actions are also s-malleable. The class of Gaussian actions has a richstructure, owing to the fact the every Gaussian action of a group Γ arises functoriallyfrom an orthogonal representation of Γ. The interplay between the representationtheory and the ergodic theory of a group via the Gaussian action has been fruitfullyexploited in the literature (cf. the seminal works of Connes & Weiss and of Schmidt,[10, 45, 46], inter alios ).In this paper, we will explore U fin -cocycle superrigidity within the class of Gauss-ian actions. An advantage to our approach is that we develop a general frameworkfor investigating cocycle superrigidity of such actions by using derivations on vonNeumann algebras. The first theme we take up is the relation between the coho-mology of group representations and the cohomology of their respective Gaussianactions. Under general assumptions, we show that cohomological information com-ing from the representation can be faithfully transferred to the cohomology groupof the action with coefficients in the circle group T . As a consequence, we obtainour first result, that the cohomology of the representation provides an obstructionto the U fin -cocycle superrigidity of the associated Gaussian action. Theorem 0.1.
Let Γ be a countable discrete group and π : Γ → O ( K ) a weakly mix-ing orthogonal representation. A necessary condition for the corresponding Gauss-ian action to be { T } -cocycle superrigid is for H (Γ , π ) = { } . The Bernoulli shift action of a group is precisely the Gaussian action correspond-ing to the left-regular representation, and the circle group T is contained in the class U fin . When combined with Corollary 2.4 in [29] which states that for a nonamenablegroup vanishing of the first ℓ -Betti number is equivalent to H (Γ , λ ) = { } we ob-tain the following corollary. Corollary 0.2.
Let Γ be a countable discrete group. If β (2)1 (Γ) = 0 then theBernoulli shift action is not U fin -cocycle superrigid. The second theme explored is the deformation/derivation duality developed bythe first author in [27]. The flexibility inherent at the infinitesimal level allows us
N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 3 to offer a unified treatment of Popa’s theorem in the case of generalized Bernoulliactions and expand the class of groups whose Bernoulli actions are known to be U fin -cocycle superrigid. As a partial converse to the above results, we have thatan a priori stronger property than having β (2)1 (Γ) = 0, L -rigidity (see Definition1.12), is sufficient to guarantee U fin -cocycle superrigidity of the Bernoulli shift. Theorem 0.3.
Let Γ be a countable discrete group. If L Γ is L -rigid then theBernoulli shift action of Γ is U fin -cocycle superrigid. Examples of groups for which this holds are groups which contain an infinitenormal subgroup which has relative property (T) or is the direct product of aninfinite group and a nonamenable group, recovering Popa’s Cocycle SuperrigidityTheorem for Bernoulli actions of these groups.We also obtain new groups for which Popa’s theorem holds. For example, weshow that the theorem holds for any generalized wreath product A ≀ X Γ , where A is a non-trivial abelian group and Γ does not have the Haagerup property. Also,if L Λ is nonamenable and has property (Γ) of Murray and von Neumann [20] thenthe theorem also holds for Λ.We remark that it is still an open question whether vanishing of the first ℓ -Bettinumber characterizes groups whose Bernoulli actions are U fin -cocycle superrigid.For instance, it is still unknown for the group Z ≀ F , which contains an infinitenormal abelian subgroup and hence has vanishing first ℓ -Betti number by [5].The authors would like to thank Sorin Popa for useful discussions regarding thiswork. 1. Preliminaries
We begin by reviewing of the basic notions of Gaussian actions, cohomology ofrepresentations and actions, and closable derivations. Though our treatment ofthe last two topics is standard, our approach to Gaussian actions is somewhat non-standard, where we take a more operator-algebraic approach by viewing the algebraof bounded functions on the probability space as a von Neumann algebra actingon a symmetric Fock space. In the noncommutative setting of free probability thisis the same as Voiculescu’s approach in [49]. But first, let us recall a few basicdefinitions and concepts which constitute the basic language in which this paper iswritten. Throughout, all Hilbert spaces are assumed to be separable.
Definition 1.1.
Let π : Γ → U ( H ) be a unitary representation, and denote by π op the associated contragradient representation on the contragradient Hilbert space H op of H . We say that π :(1) is ergodic if π has no non-zero invariant vectors;(2) is weakly mixing if π ⊗ π op is ergodic (equivalently, π ⊗ ρ op is ergodic forany unitary Γ-representation ρ );(3) is mixing if h π γ ( ξ ) , η i → γ → ∞ , for all ξ, η ∈ H ;(4) has spectral gap if there exists K ⊂ G , finite, and C > k ξ − P ( ξ ) k ≤ C P k ∈ K k π k ( ξ ) − ξ k , for all ξ ∈ H , where P is the projectiononto the invariant vectors;(5) has stable spectral gap if π ⊗ π op has spectral gap (equivalently, π ⊗ ρ op has spectral gap for any unitary Γ-representation ρ );(6) is amenable if π is either not weakly mixing or does not have stable spectralgap. JESSE PETERSON AND THOMAS SINCLAIR
Note that for an orthogonal representation π of Γ into a real Hilbert space K ,the associated unitary representation into K ⊗ C is canonically isomorphic to itscontragradient. Hence, in this situation we may replace in the above definition“ π ⊗ π op ” and “ π ⊗ ρ op ” with “ π ⊗ π ” and “ π ⊗ ρ ”, respectively.Let Γ y σ ( X, µ ) be an action of the countable discrete group Γ by µ -preservingautomorphisms of a standard probability space ( X, µ ). This yields a unitary repre-sentation π σ : Γ → U ( L ( X, µ )) called the
Koopman representation associatedto σ . (Here L ( X, µ ) denotes the orthogonal complement in L ( X, µ ) to the sub-space of the constant functions on X .) Note that the Koopman representation isthe unitary representation associated to the orthogonal representation of Γ actingon the real-valued L -functions. We say that the action σ is ergodic (or weaklymixing, mixing, etc.) if the Koopman representation π σ is in the sense of the abovedefinition. An action Γ y σ ( X, µ ) is (essentially) free if, for all γ ∈ Γ, γ = e , µ { x ∈ X : σ γ ( x ) = x } = 0.Given unitary representations π : Γ → U ( H ) and ρ : Γ → U ( K ), we say that π is contained in ρ if there is a linear isometry V : H → K such that π γ = V ∗ ρ γ V ,for all γ ∈ Γ. We say that π is weakly contained ρ if for any ξ, η ∈ H , thereare ξ ′ n , η ′ n ∈ K such that h π γ ( ξ ) , η i = Σ n ∈ N h ρ γ ( ξ ′ n ) , η ′ n i , for all γ ∈ Γ. Note thatamenability of a representation π is equivalent to π ⊗ π op weakly containing the triv-ial representation, which is equivalent with Bekka’s definition by Theorem 5.1 in [1].The “representation theory” of a II factor is captured in the structure of itsbimodules, also called correspondences (cf. [30]). The theory of correspondenceswas first developed by A. Connes [8]. Definition 1.2.
Let M be a II factor. An M - M Hilbert bimodule is a Hilbertspace H equipped with a representation π : M ⊗ alg M op → B ( H ) which is normalwhen restricted to M and M op . We write π ( x ⊗ y op ) ξ as xξy .An M - M Hilbert bimodule H is contained in an M - M Hilbert bimodule K ifthere is a linear isometry V : H → K such that V ( xξy ) = xV ( ξ ) y , for all ξ ∈ H , x, y ∈ M ; H is weakly contained in K if for any ξ, η ∈ H , F ⊂ M finite, thereexist ξ ′ n , η ′ n ∈ K such that h xξy, η i = Σ n ∈ N h xξ ′ n y, η ′ n i , for all x, y ∈ F . The trivialbimodule is the space L ( M, τ ) with the obvious bimodule structure induced byleft and right multiplication; the coarse bimodule is the space L ( M, τ ) ⊗ L ( M, τ )with bimodule structure induced by left multiplication on the first factor and rightmultiplication on the second. The trivial and coarse bimodules play analogous rolesin the theory of M - M Hilbert bimodules to the roles played, respectively, by thetrivial and left-regular representations in the theory of unitary representations oflocally-compact groups. Note that an M - M correspondence H contains the trivialcorrespondence if and only if H has non-zero M -central vectors (a vector ξ is M -central if xξ = ξx , for all x ∈ M ).Given ξ, η ∈ H , note the maps M ∋ x
7→ h xξ, η i , h ξx, η i are normal linearfunctionals on M . A vector ξ ∈ H is called left (respectively, right) bounded if there exists C > x ∈ M , k xξ k ≤ C k x k , (resp., k ξx k ≤ C k x k ). The set of vectors which are both left and right-bounded forms a densesubspace of H . By [30], to ξ , a left-bounded vector, we can associate a completely-positive map φ ξ : M → M such that for all x, y ∈ M , k xξy k = τ ( x ∗ xφ ξ ( yy ∗ )) / .If ξ is also right-bounded then this map is seen to naturally extend to a boundedoperator ˆ φ ξ : L ( M, τ ) → L ( M, τ ). N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 5
Given two M - M Hilbert bimodules H and K , there is a well-defined tensorproduct H ⊗ M K in the category of M - M Hilbert bimodules: see [30] for details.
Definition 1.3 (Compare with Definition 1.1.) . Let H be an M - M Hilbert bimod-ule. H is said to:(1) be weakly mixing if H ⊗ M H op does not contain the trivial M - M Hilbertbimodule;(2) be compact (or mixing) if for every sequence u i ∈ U ( M ) such that u i →
0, weakly, we have thatlim i →∞ sup k x k≤ h u i ξx, η i = lim i →∞ sup k x k≤ h xξu i , η i = 0 , for all ξ, η ∈ H (equivalently, ˆ φ ξ is a compact operator on L ( M, τ ) forevery left-bounded vector ξ ∈ H );(3) have spectral gap if there exist x , . . . , x n ∈ M such that k ξ − P ( ξ ) k ≤ P ni =1 k x i ξ − ξx i k , for all ξ ∈ H , where P is the projection onto the centralvectors;(4) have stable spectral gap if H ⊗ M H op has spectral gap;(5) be amenable if it is either not weakly mixing or does not have stablespectral gap.If H is a compact M -correspondence and K an arbitary M -correspondence, then H ⊗ M K (and also K ⊗ M H ) is compact, since ˆ φ ξ ⊗ M η = ˆ φ η ◦ ˆ φ ξ if ξ and η are bothleft and right-bounded.Let H and K be M - M correspondences, and denote by H and K the set ofright-bounded vectors in H and K , respectively. For ξ, η ∈ H , denote by ( ξ | η ) theelement of M such that h ξx, ηy i = τ ( y ∗ ( ξ | η ) x ), for all x, y ∈ X (by normality ofthe map xy ∗
7→ h ξxy ∗ , η i , there exists such a ( ξ | η ) ∈ L ( M, τ ); right-boundednessof ξ and η implies ( ξ | η ) ∈ M ). It is clear that ( ·|· ) is a bilinear map H × H → M such that ( ξ | ξ ) ≥ ξ | ξ ) = 0 if and only if ξ = 0, for all ξ ∈ H . For ξ ∈ H and η ∈ K , define the linear map T ξ,η : H → K op by T ξ,η ( · ) = ( ·| ξ ) η op . It is easyto check that T ξ,η is a bounded with k T ξ,η k = k ξ k − / k ( ξ | ξ ) η k ; hence, T ξ,η extendsto a bounded operator H → K op . Let L M ( H , K ) be the subspace of B ( H , K op )which is the closed span of all operators of the form T ξ,η under the Hilbert norm k T ξ,η k L M = τ (( ξ | ξ )( η | η )) / . Moreover, L M ( H , K ) is equipped with a natural M - M Hilbert bimodule structure given by ( x ⊗ y op )( T ξ,η ) = T xξ,yη identifying it with H ⊗ M K op . Note that if T ∈ L M ( H , K ), then ( T ∗ T ) / ∈ L M ( H , H ). Proposition 1.4. An M - M correspondence H is weakly mixing if and only if forany M - M correspondence K , H⊗ M K op does not contain the trivial correspondence. Proof.
The reverse implication is trivial. Conversely, suppose there exists K suchthat H ⊗ K op contains an M -central vector. Identifying H ⊗ M K op with L M ( H , K ),let T ∈ L M ( H , K ) be an M -central vector. Then ( T ∗ T ) / ∈ L M ( H , H ) is an M -central vector; hence, H is not weakly mixing. (cid:3) Gaussian actions.
Let π : Γ → O ( H ) be an orthogonal representation of acountable discrete group Γ. The aim of this section is to describe the constructionof a measure-preserving action of Γ on a non-atomic standard probability space( X, µ ) such that H is realized as a subspace of L R ( X, µ ) and π is contained in JESSE PETERSON AND THOMAS SINCLAIR the Koopman representation Γ y L ( X, µ ). The action Γ y ( X, µ ) is referred toas the
Gaussian action associated to π . We give an operator-algebraic alterna-tive construction of the Gaussian action similar to Voiculescu’s construction of freesemi-circular random variables.Given a real Hilbert space H , the n-symmetric tensor H ⊙ n is the subspace of H ⊗ n fixed by the action of the symmetric group S n by permuting the indices. For ξ , . . . , ξ n ∈ H , we define their symmetric tensor product ξ ⊙ · · · ⊙ ξ n ∈ H ⊙ n to be n ! P σ ∈ S n ξ σ (1) ⊗ · · · ⊗ ξ σ ( n ) . Denote S ( H ) = C Ω ⊕ ∞ M n =1 ( H ⊗ C ) ⊙ n , with renormalized inner product such that k ξ k S ( H ) = n ! k ξ k , for ξ ∈ H ⊙ n .For ξ ∈ H let x ξ be the symmetric creation operator , x ξ (Ω) = ξ, x ξ ( η ⊙ · · · ⊙ η k ) = ξ ⊙ η ⊙ · · · ⊙ η k , and its adjoint, ∂∂ξ = ( x ξ ) ∗ ∂∂ξ (Ω) = 0 , ∂∂ξ ( η ⊙ · · · ⊙ η k ) = k X i =1 h ξ, η i i η ⊙ · · · ⊙ b η i ⊙ · · · ⊙ η k . Let s ( ξ ) = 12 ( x ξ + ∂∂ξ ) , and note that it is an unbounded, self-adjoint operator on S ( H ).The moment generating function M ( t ) for the Gaussian distribution is de-fined to be M ( t ) = 1 √ π Z ∞−∞ exp( tx ) exp( − x / dx = exp( t / . It is easy to check that if k ξ k = 1 then h s ( ξ ) n Ω , Ω i = M ( n ) (0) = (2 k )!2 k k ! , if n = 2 k and 0 if n is odd. Hence, s ( ξ ) may be regarded as a Gaussian randomvariable. Note that if ξ, η ∈ H then s ( ξ ) and s ( η ) commute, moreover, if ξ ⊥ η ,then h s ( ξ ) m s ( η ) n Ω , Ω i = 0, for all m, n ∈ N ; thus, s ( ξ ) and s ( η ) are independentrandom variables.From now on we will use the convention ξ ξ . . . ξ k to denote the symmetrictensor ξ ⊙ ξ ⊙ · · · ⊙ ξ k . Let Ξ be a basis for H and S (Ξ) = { Ω } ∪ { s ( ξ ) s ( ξ ) . . . s ( ξ k )Ω : ξ , ξ , . . . , ξ k ∈ Ξ } . Lemma 1.5.
The set S (Ξ) is a (non-orthonormal) basis of S ( H ) .Proof. We will show that ξ . . . ξ k ∈ span( S (Ξ)), for all ξ , . . . , ξ k ∈ H . We haveΩ ∈ span( S (Ξ)). Also, since s ( ξ )Ω = ξ , H ⊂ span( S (Ξ)). Now as s ( ξ ) . . . s ( ξ k )Ω = P ( ξ , . . . , ξ k ) is a polynomial in ξ , . . . , ξ k of degree k with top term ξ . . . ξ k , theresult follows by induction on k . (cid:3) N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 7
Let u ( ξ , . . . , ξ k ) = exp( πis ( ξ ) . . . s ( ξ k )) and u ( ξ , . . . , ξ k ) t = exp( πits ( ξ ) . . . s ( ξ k )).Denote by A the von Neumann algebra generated by all such u ( ξ , . . . , ξ k ), whichis the same as the von Neumann algebra generated by the spectral projections ofthe unbounded operators s ( ξ ) . . . s ( ξ k ). Theorem 1.6.
We have that L ( A, τ ) ∼ = S ( H ) , and A is a maximal abelian ∗ -subalgebra of B ( S ( H )) with faithful trace τ = h· Ω , Ω i . In particular, A is a diffuseabelian von Neumann algebra.Proof. By Lemma 1.5, A A Ω is an embedding of A into S ( H ). By Stone’sTheorem lim t → u ( ξ , . . . , ξ k ) t − πit Ω = s ( ξ ) . . . s ( ξ k )Ω;hence, A Ω is dense in S ( H ). This implies that A is maximal abelian in B ( S ( H )). (cid:3) There is a natural strongly-continuous embedding O ( H ) ֒ → U ( S ( H )) given by T T S = 1 ⊕ ∞ M n =1 T ⊙ n . It follows that there is an embedding O ( H ) → Aut(
A, τ ), T σ T , which can beidentified on the unitaries u ( ξ , . . . , ξ k ) by σ T ( u ( ξ , . . . , ξ k )) = Ad( T S )( u ( ξ , . . . , ξ k )) = u ( T ( ξ ) , . . . , T ( ξ k )) . Thus for an orthogonal representation π : Γ → O ( H ), there is a natural ac-tion σ π : Γ → Aut(
A, τ ) given by σ πγ ( u ( ξ , . . . , ξ k )) = u ( π γ ( ξ ) , . . . , π γ ( ξ k )) =Ad( π S γ )( u ( ξ , . . . , ξ k )). The action σ π is the Gaussian action associated to π .We have that ergodic properties which remain stable with respect to tensorproducts tranfer from π to σ π . Proposition 1.7.
In particular, for a subgroup H ≤ Γ, σ π | H possesses any of thefollowing properties if and only if π | H does:(1) weak mixing;(2) mixing;(3) stable spectral gap;(4) being contained in a direct sum of copies of the left-regular representation;(5) being weakly contained in the left-regular representation.For Gaussian actions, stable properties are equivalent to their “non-stable” coun-terparts. The following proposition serves as a prototype of such a result, showingthat ergodicity implies stable ergodicity, i.e. , weak mixing. Theorem 1.8. Γ y σ π ( A, τ ) is ergodic if and only if π is weakly mixing.Proof. The reverse implication follows from Proposition 1.7. Conversely, supposethere exists ξ ∈ H ⊗ such that for all γ ∈ Γ, π γ ( ξ ) = ξ . Viewing ξ as a Hilbert-Schmidt operator on H , let | ξ | = ( ξξ ∗ ) / . Since the map ξ ⊗ η η ⊗ ξ is the sameas taking the adjoint of the corresponding Hilbert-Schmidt operator, we have that | ξ | ∈ H ⊙ and π γ ( | ξ | ) = | ξ | . By functional calculus, there exists λ >
0, such that η = E λ ( | ξ | ) = 0 is a finite rank operator. Hence, η = η ⊙ η + · · · + η n ⊙ η n ∈ H ⊙ with η i ⊙ η i ⊥ η j ⊙ η j for i = j . But then u = Q ni =1 u ( η i , η i ) ∈ A , a non-trivialunitary and σ πγ ( u ) = u . Hence, σ π is not ergodic. (cid:3) JESSE PETERSON AND THOMAS SINCLAIR
Cohomology of a representation & action.
Let K be a real Hilbert spaceand π : Γ → O ( K ) an orthogonal representation of a countable discrete group Γ. Definition 1.9. A cocycle is a map b : Γ → K satisfying the cocycle identity b ( γ γ ) = π γ b ( γ ) + b ( γ ), for all γ , γ ∈ Γ. A cocycle is a coboundary is thereexists η ∈ K such that b ( γ ) = π γ η − η , for all γ ∈ Γ.It is a well-known fact (cf. [2]) that a cocycle b is a coboundary if and onlyif sup γ ∈ Γ k b ( γ ) k < ∞ . Let Z (Γ , π ) and B (Γ , π ) denote, respectively, the vectorspace of all cocycles and the subspace of coboundaries. The first cohomologyspace H (Γ , π ) of the representation π is then defined to be Z (Γ , π ) /B (Γ , π ).Let Γ y σ ( X, µ ) be an ergodic, measure-preserving action on a standard proba-bility space (
X, µ ), and let A be a Polish topological group. Definition 1.10. A cocycle is a measurable map c : Γ × X → A satisfying thecocycle identity c ( γ γ , x ) = c ( γ , σ γ ( x )) c ( γ , x ), for all γ , γ ∈ Γ, a . e . x ∈ X .A pair of cocycles c , c are cohomologous (written c ∼ c ) if there exists ameasurable map ξ : X → A such that ξ ( σ γ ( x )) c ( γ, x ) ξ ( x ) − = c ( γ, x ) for all γ ∈ Γ, a . e . x ∈ X . A cocycle is a coboundary if it is cohomologous to the cocyclewhich is identically 1.Let Z (Γ , σ, A ) and B (Γ , σ, A ) denote, respectively, the space of all cocycles andthe subspace of coboundaries. The first cohomology space H (Γ , σ, A ) of theaction σ is defined to be Z (Γ , σ, A ) / ∼ . Note that if A is abelian, Z (Γ , σ, A ) is en-dowed with a natural abelian group structure and H (Γ , σ, A ) = Z (Γ , σ, A ) /B (Γ , σ, A ).To any homomorphism ρ : Γ → A we can associate a cocycle ˜ ρ by ˜ ρ ( γ, x ) = ρ ( γ ).Using terminology developed by Popa (cf. [36]), a cocycle c is said to untwist ifthere exists a homomorphism ρ : Γ → A such that c is cohomologous to ˜ ρ . Toany cocycle c ∈ Z (Γ , σ, A ), we can associated two cocycles c ℓ , c r ∈ Z (Γ , σ × σ, A )given by c ℓ ( γ, ( x, y )) = c ( γ, x ) and c r ( γ, ( x, y )) = c ( γ, y ). It is easy to check that c untwists only if c ℓ is cohomologous to c r ; if σ is weakly mixing, Theorem 3.1 in[36] establishes the converse.1.3. Closable derivations.
We review here briefly some general properties of clos-able derivations on a finite von Neumann algebra and set up some notation to beused in the sequel. For a more detailed discussion see [12], [26], [27], or [24].
Definition 1.11.
Let (
N, τ ) be a finite von Neumann algebra and H be an N - N correspondence. A derivation δ is an unbounded operator δ : L ( N, τ ) → H suchthat D ( δ ) is a k · k -dense ∗ -subalgebra of N , and δ ( xy ) = xδ ( y ) + δ ( x ) y , for each x, y ∈ D ( δ ). A derivation is closable if it is closable as an operator and real if H has an antilinear involution J such that J ( xξy ) = y ∗ J ( ξ ) x ∗ , and J ( δ ( z )) = δ ( z ∗ ),for each x, y ∈ N , ξ ∈ H , z ∈ D ( δ ).If δ is a closable derivation then by [12] D ( δ ) ∩ N is again a ∗ -subalgebra and δ | D ( δ ∩ N ) is again a derivation. We will thus use the slight abuse of notation bysaying that δ is a closed derivation.To every closed real derivation δ : N → H , we can associate a semigroup defor-mation Φ t = exp( − tδ ∗ δ ), t >
0, and a resolvent deformation ζ α = ( α/ ( α + δ ∗ δ )) / , α >
0. Both of these deformations are of unital, symmetric, completely-positivemaps; moreover, the derivation δ can be recovered from these deformations [43, 44]. N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 9
We also have that the deformation Φ t converges uniformly on ( N ) as t → ζ α converges uniformly on ( N ) as α → ∞ . Definition 1.12 (Definition 4.1 in [27]) . Let (
N, τ ) be a finite von Neumannalgebra. N is L -rigid if given any inclusion ( N, τ ) ⊂ ( M, ˜ τ ), and any closable realderivation δ : M → H such that H when viewed as an N - N correspondence embedsin ( L N ⊗ L N ) ⊕∞ , we then have that the associated deformation ζ α convergesuniformly to the identity in k · k on the unit ball of N .We point out here that our definition above is formally stronger than the onegiven in [27]. Specifically, there it was assumed that H embedded into the coarsebimodule as an M - M bimodule rather than an N - N bimodule. However, this extracondition was not used in [27], and since the above definition has better stabilityproperties (see Theorem 5.3) we have chosen to use the same terminology.Examples of nonamenable groups which do not give rise to L -rigid group vonNeumann algebras are groups such that the first ℓ -Betti number is positive. Theseare, in fact, the only known examples, and L -rigidity should be viewed as a vonNeumann analog of vanishing first ℓ -Betti number.Showing that a group von Neumann algebra is L -rigid can be quite difficult ingeneral since one has to consider derivations which may not be defined on the groupalgebra. Nonetheless, there are certain situations where this can be verified. Theorem 1.13 (Corollary 4.6 in [27]) . Let Γ be a nonamenable countable discretegroup. If L Γ is weakly rigid, non-prime, or has property (Γ) of Murray and vonNeumann, then L Γ is L -rigid. We give another class of examples below (see also [23], [24], or [28]). The gapbetween group von Neumann algebras which are known to be L -rigid and groupswith vanishing first ℓ -Betti number is, however, quite large. For example, as wementioned in the introduction, the wreath product Z ≀ F is a group which hasvanishing first ℓ -Betti number but for which it is not known whether the groupvon Neumann algebra is L -rigid.2. Deformations
In this section and Section 4 we will discuss the interplay between one-parametergroups of automorphisms or, more generally, semigroups of completely-positivemaps of finite factors (deformations) and their infinitesimal generators (deriva-tions). The motivation for studying deformations at the infinitesimal level is thatit allows for the creation of other related deformations of the algebra. And whilePopa’s deformation/rigidity machinery requires uniform convergence of the origi-nal deformation on some target subalgebra, it is often more feasible to demonstrateuniform convergence of a related deformation, then transfer those estimates backthe the original.We begin by recalling Popa’s notion of an s-malleable deformation, and givesome examples of such deformations that have appeared in the literature.
Definition 2.1 (Definition 4.3 in [36]) . Let (
M, τ ) be a finite von Neumann algebrasuch that (
M, τ ) ⊂ ( ˜ M , ˜ τ ). A pair ( α, β ), consisting of a point-wise strongly con-tinuous one-parameter family α : R → Aut( ˜
M , ˜ τ ) and an involution β ∈ Aut( ˜
M , ˜ τ )is called a s-malleable deformation of M if:(1) M ⊂ ˜ M β ; (2) α t ◦ β = β ◦ α − t ; and(3) α ( M ) ⊥ M .2.1. Popa’s deformation.
The following deformation was used by Popa in [36]to obtain cocycle superrigidity for generalized Bernoulli actions of property (T)groups.Let (
A, τ ) be a finite diffuse abelian von Neumann algebra and u, v ∈ A ⊗ A begenerating Haar unitaries for A ⊗ , ⊗ A ⊂ A ⊗ A , respectively. Set w = u ∗ v .Choose h ∈ A ⊗ A self-adjoint such that exp( πih ) = w , and let w t = exp( πith ).Since { w } ′′ ⊥ A ⊗ , ⊗ A , we have that for any t , w t u and w t v are again Haarunitaries. Moreover, since w ∈ { w t u, w t v } ′′ , { w t u, w t v } is a pair of generating Haarunitaries in A ⊗ A . Hence there is a well-defined one-parameter family α : R → Aut( A ⊗ A, τ ⊗ τ ) given by α t ( u ) = w t u, α t ( v ) = w t v. The family α , together with the automorphism β given by β ( u ) = u, β ( v ) = u v ∗ , is seen to be an s-malleable deformation of A ⊗ ⊂ A ⊗ A . Definition 2.2.
Let (
P, τ ) be a finite von Neumann algebra and σ : Γ → Aut(
P, τ )a Γ-action. Γ y σ ( P, τ ) is an s-malleable action if there exists an s-malleabledeformation ( α, β ) of (
P, τ ) such that β, α t commute with σ γ ⊗ σ γ for all t ∈ R , γ ∈ Γ. For any countable discrete group there is a canonical example of an s-malleableaction, the
Bernoulli shift . Let (
A, τ ) = ( L ∞ ( T , λ ) , R · dλ ), ( X, µ ) = Q g ∈ Γ ( T , λ ),and ( B, τ ′ ) = N γ ∈ Γ ( A, τ ). The Bernoulli shift is the natural action Γ y σ ( X, µ )defined by shifting indices: σ γ (( x γ ) γ ) = ( x γ ) γ γ = ( x γ − γ ) γ . Defining˜ α t ((˜ x γ ) γ ) = ( α t (˜ x γ )) γ and ˜ β ((˜ x γ ) γ ) = ( β (˜ x γ )) γ , for (˜ x γ ) γ ∈ ˜ B = N γ ∈ Γ ( A ⊗ A ) ∼ = B ⊗ B , we see that (˜ α, ˜ β ) is an s-malleabledeformation of B which commutes with the Bernoulli Γ-action.2.2. Ioana’s deformation.
The deformation described below was first used byIoana [15] in the case when the base space is nonamenable, and later used byChifan and Ioana [6] in part to obtain solidity of L ∞ ( X, µ ) ⋊ σ Γ, whenever L Γ solidand Γ y σ ( X, µ ), the Bernoulli shift. Their deformation is inspired by the freeproduct deformation used in [16]. A similar deformation has also been previouslyused by Voiculescu in [51].Given a finite von Neumann algebra (
B, τ ), let ˜ B = B ∗ L ( Z ). If u ∈ U ( L ( Z ))is a generating Haar unitary, choose an h ∈ L ( Z ) such that exp( πih ) = u , and let u t = exp( πith ). Define the deformation α : R → Aut( ˜ B, ˜ τ ) by α t = Ad( u t ) . Let β ∈ Aut( ˜ B, ˜ τ ) be defined by β | B = id and β ( u ) = u ∗ . It is easy to check that ( α, β ) is a s-malleable deformation of B . N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 11
If a countable discrete group Γ acts on a countable set S then we may considerthe generalized Bernoulli shift action of Γ on ⊗ s ∈ S B given by σ γ ( ⊗ s ∈ S b s ) = ⊗ s ∈ S b γ − s . We then have that ⊗ s ∈ S B ⊂ ⊗ s ∈ S ˜ B and ( ⊗ s ∈ S α, ⊗ s ∈ S β ) gives as-malleable deformation of ⊗ s ∈ S B .2.3. Malleable deformations of Gaussian actions.
We will now construct thecanonical s-malleable deformation of a Gaussian action which is given in Section4.3 of [17], and give an explicit description of its associated derivation. To begin,let π : Γ → O ( H ) be an orthogonal representation, ˜ H = H ⊕ H , and ˜ π = π ⊕ π .If σ π : Γ → Aut(
A, τ ) is the Gaussian action associated with π , then the Gaussianaction associated to ˜ π is naturally identified with the action σ π ⊗ σ π on A ⊗ A . Let˜ σ π = σ π ⊗ σ π .Let J = (cid:0) − (cid:1) , the operator which gives ˜ H the structure of a complexHilbert space, and consider the one-parameter unitary group θ t = exp( πt J ). Since θ t commutes with ˜ π , there is a well-defined one-parameter group α : R → Aut( A ⊗ A, τ ⊗ τ ) which commutes with ˜ σ π namely, α t = σ θ t = Ad(exp( πt J ) S ) . Let ρ = (cid:0) − (cid:1) , and observe ρ ◦ θ − t = θ t ◦ ρ . Hence, β = σ ρ = Ad( ρ S )conjugates α t and α − t . Finally notice that θ ( H ⊕
0) = 0 ⊕ H , which gives α ( A ⊗
1) = 1 ⊗ A . The pair ( α, β ) is, thusly, an s-malleable deformation ofthe action σ π .Let T ∈ B ( ˜ H ) be skew-adjoint. Associate to T the unbounded skew-adjointoperator ∂ ( T ) on S ( H ) defined by ∂ ( T )(Ω) = 0 , ∂ ( T )( ξ . . . ξ n ) = n X i =1 ξ . . . T ( ξ i ) . . . ξ n . We have that if U ( t ) = exp( tT ) ∈ O ( H ), thenlim t → U ( t ) S − It = ∂ ( T ) . Let δ : A ⊗ A → L ( A ⊗ A ) be the derivation defined by δ ( x ) = [ x, ∂ ( T )] = lim t → σ U ( t ) ( x ) − xt . Taking T to be the operator J defined above, gives us the derivation which is theinfinitesimal generator of the s-malleable deformation of the Gaussian action de-scribed in this section. From the relation δ ( · ) = [ · , ∂ ( J )], we see that the ∗ -algebragenerated by the operators s ( ξ ) forms a core for δ .Letting δ = δ | A ⊗ , we have thatΦ t = exp( − tδ ∗ δ ) = exp( − tE A ⊗ ◦ δ ∗ δ ) = exp( tE A ⊗ ◦ δ ) . We compute E A ⊗ ◦ δ ( s ( ξ ) . . . s ( ξ k )) = − k s ( ξ ) . . . s ( ξ k ) . Hence, Φ t ( s ( ξ ) . . . s ( ξ k )) = (1 − e − kt ) s (Ω) + e − kt s ( ξ ) . . . s ( ξ k ) . Cohomology of Gaussian actions
In this section, we obtain Theorem 0.1 and its corollary. We do so by using aconstruction (cf. [19], [25], [46]) which, given an orthogonal representation and acocycle, produces a T -valued cocycles for the associated Gaussian action. We thenshow that these cocycles do not untwist by applying the above deformation.Let b : Γ → H be a cocycle for an orthogonal representation π : Γ → O ( H )and Γ y σ ( A, τ ) = ( L ∞ ( X, µ ) , R · dµ ) be the Gaussian action associated to π , asdescribed in section 1.1. Viewing H as a subset of L R ( X, µ ), Parthasarathy andSchmidt in [25] constructed the cocycle c : Γ × X → T by the rule c ( γ, x ) = exp( ib ( γ − ))( x ) . We write ω γ for the element of U ( L ∞ ( X, µ )) given by ω γ ( x ) = c ( γ, γ − x ). Thecocycle identity for c then transforms to the formula ω γ γ = ω γ σ γ ( ω γ ), for all γ , γ ∈ Γ. Moreover, c is cohomologous to a homomorphism if and only if there isa unitary element u ∈ U ( L ∞ ( X, µ )) such that γ uω γ σ γ ( u ∗ ) is a homomorphism,i.e., each uω γ σ γ ( u ∗ ) is fixed by the action of the group.A routine calculation shows that τ ( ω γ ) = R c ( γ, x ) dµ ( x ) = exp( −k b ( γ ) k / ϕ ( γ ) = exp( −k b ( γ ) k /
2) is naturally isomorphic to the twisted Gaussianaction ω γ σ γ . Theorem 3.1.
Using the notation above, if π : Γ → O ( H ) is weak mixing, (so that σ is ergodic) and if b is an unbounded cocycle, then c does not untwist.Proof. Since σ is ergodic, if c were to untwist then there would exist some u ∈ U ( A )such that uω γ σ γ ( u ) ∈ T , for all γ ∈ Γ. It would then follow that any deformationof A which commutes with the action of Γ must converge uniformly on the set { ω γ | γ ∈ Γ } . Indeed, this is just a consequence of the fact that completely positivedeformations become asymptotically A -bimodular.However, if we apply the deformation α t from Section 2.3 then we can compute h α t/π ( ω γ ⊗ , ω γ ⊗ i = h exp( i (cos t ) b ( γ − )) ⊗ exp( − i (sin t ) b ( γ − )) , exp( ib ( γ − )) ⊗ i = exp((1 − cos t ) k b ( γ ) k / t ) k b ( γ ) k / − (1 − cos t ) k b ( γ ) k )This will converge uniformly for γ ∈ Γ if and only if the cocycle b is boundedand hence the result follows. (cid:3) Corollary 3.2.
The exponentiation map described above induces an injective ho-momorphism H (Γ , π ) → H (Γ , σ, T ) /χ (Γ) , where χ (Γ) is the character group of Γ . N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 13
Proof.
It is easy to see that if two cocycles in Z (Γ , π ) are cohomologous then theresulting cocycles for the Gaussian action will also be cohomologous. This showsthat the map described above is well defined.The above theorem, together with the fact that this map is a homomorphism,shows that this map is injective. (cid:3) Since a nonamenable group has vanishing first ℓ -Betti number if and only if ithas vanishing first cohomology into its left regular representation [3], [29], we derivethe following corollary. Corollary 3.3.
Let Γ be a nonamenable countable discrete group, and let Γ y σ ( X, µ ) be the Bernoulli shift action. If β (2)1 (Γ) = 0 then H (Γ , σ, T ) = χ (Γ) , where χ (Γ) is the group of characters. In particular, Γ y σ ( X, µ ) is not U fin -cocyclesuperrigid. Derivations
In this section we continue our investigation of deformations, but this time onthe infinitesimal level.4.1.
Derivations from s-malleable deformations.
Let (
M, τ ) be a finite vonNeumann algebra, and let α : R → Aut(
M, τ ) be a point-wise strongly continuousone-parameter group of automorphisms. Let δ be the infinitesimal generator of α , i.e. , exp( tδ ) = α t . For f ∈ L ( R ) define the bounded operator α f : M → M by α f ( x ) = Z ∞−∞ f ( s ) α s ( x ) ds. It can be checked that if f ∈ C ( R ) ∩ L ( R ) and f ′ ∈ L ( R ), then δ ◦ α f ( x ) = − α f ′ ( x ) . Also if x ∈ M ∩ D ( δ ), then we have that α t ( x ) − x = Z t δ ◦ α s ( x ) ds = Z t α s ( δ ( x )) ds. Theorem 4.1.
Suppose that for every ε > , there exists f ∈ C ( R ) ∩ L ( R ) suchthat f ′ ∈ L ( R ) and sup x ∈ ( M ) k α f ( x ) − x k ≤ ε/ . Then α t converges k · k -uniformly to the identity on ( M ) as t → .Proof. We need only show for every ε > η > t < η , sup x ∈ ( M ) k α t ( x ) − x k ≤ ε . Let ˜ x = α f ( x ). We have that k α t ( x ) − x k ≤ k α t (˜ x ) − ˜ x k + ε/
2. Since δ ◦ α f is defined everywhere, δ ◦ α f : M → L ( M, τ ) is bounded. In fact, k δ ◦ α f k ≤ k f ′ k L . Now, since ˜ x ∈ D ( δ ),we have α t (˜ x ) − ˜ x = R t α s ( δ (˜ x )) ds . Hence k α t (˜ x ) − ˜ x k ≤ t k f ′ k L . Choosing η = ε (2 k f ′ k L ) − does the job. (cid:3) Corollary 4.2. If ϕ t = exp( − tδ ∗ δ ) converges uniformly to the identity as t → ,then so does α t .Proof. Let f t ( s ) = √ πt exp( − s / t ); then, ϕ t ( x ) = R ∞−∞ f t ( s ) α s ( x ) ds follows bycompleting the square. (cid:3) Tensor products of derivations.
We describe here the notion of a tensorproduct of derivations; see also Section 6 of [27].Consider N i , i ∈ I a family of finite von Neumann algebras with normal faithfultraces τ i . If δ i : N i → H i is a family of closable real derivations into Hilbertbimodules H i with domains D ( δ i ) then we may consider the dense ∗ -subalgebra D ( δ ) = ⊗ alg i ∈ I D ( δ i ) ⊂ N = ⊗ i ∈ I N i .We denote by ˆ N j the tensor product of the N i ’s obtained by omitting the j thindex so that we have a natural identification N = ˆ N j ⊗ N j for each j ∈ I . Let H = L j ∈ I H j ⊗ L ( ˆ N j ) which is naturally a Hilbert bimodule because of the iden-tification N = ˆ N j ⊗ N j .The tensor product of the derivations δ i , i ∈ I is defined to be the derivation δ = N i ∈ I δ i : D ( δ ) → H which satisfies δ ( ⊗ i ∈ I x i ) = M j ∈ I ( δ j ( x j ) ⊗ i ∈ I,i = j x i ) . This is well defined as only finitely many of the x i ’s are not equal to 1 and hencethe right hand side is a finite sum.If Φ ti = exp( − tδ ∗ i δ i ) is the semigroup deformation associated to δ i then one easilychecks that the semigroup deformation associated to δ is Φ t = N i ∈ I Φ ti : N → N .A similar formula holds for the resolvent deformation. Note that by viewing theHilbert bimodule associated with Φ t and using the usual “averaging trick” (e.g.Theorem 4.2 in [30]) it follows that Φ t will converge uniformly in k · k to theidentity on ( N ) if and only if each Φ it converges uniformly in k · k to the identityon ( N i ) and moreover this convergence is uniform in i ∈ I .4.3. Derivations from generalized Bernoulli shifts.
We use here the notationin Section 1.1 above. Given a real Hilbert space H , we consider the new Hilbertspace H ′ = R Ω ⊕ H . If ξ ∈ H is a non-zero element we denote by P ξ the rankone projection onto ξ . We denote by ˜ H the tensor product (complex) Hilbert space H ⊗ S ( H ′ )Let N ∈ N ∪ {∞} be the dimension of H and consider an orthonormal basis β = { ξ n } Ni =1 for H . We then define a left action of A , the von Neumann algebragenerated by the spectral projections of s ( ξ ), ξ ∈ H , on ˜ H such that for each ξ ∈ H , s ( ξ ) acts on the left (as an unbounded operator) by ℓ β ( s ( ξ )) = id ⊗ s ( ξ ) . We also define a right action of A on ˜ H such that for each ξ ∈ H , s ( ξ ) acts on theright by extending linearly the formula(1) r β ( s ( ξ ))( ξ n ⊗ η ) = P ξ n ( ξ ) ⊗ S (Ω ) η + ξ n ⊗ s ( ξ − P ξ n ( ξ )) η, for each 1 ≤ n ≤ N, η ∈ S ( H ′ ).These formulas define unbounded self-adjoint operators on ˜ H in general; however,by functional calculus they extend to give commuting normal actions of A on ˜ H .Moreover, if T ∈ O ( H ) ⊂ O ( H ′ ), then we have that for any ξ ∈ H ℓ T β ( s ( T ξ )) = ℓ T β ( σ T ( s ( ξ ))) = Ad( T ⊗ T S ) ℓ β ( s ( ξ )) . Also, r T β ( s ( T ξ )) = r T β ( σ T ( s ( ξ ))) = Ad( T ⊗ T S )( r β ( s ( ξ ))) . N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 15
From here on we will denote the left action of A on ˜ H by ℓ β ( a ) x = a · β x and theright action by r β ( a ) x = x · β a . By extending the formulas above to A we have thefollowing lemma. Lemma 4.3.
Using the notation above, consider the inclusion O ( H ) ⊂ U ( ˜ H ) givenby T ˜ T = T ⊗ T S . Then for each T ∈ O ( H ) , x, y ∈ A , and ˜ ξ ∈ ˜ H , we have ˜ T ( x · β ˜ ξ · β y ) = σ T ( x ) · T β ( ˜ T ˜ ξ ) · T β σ T ( y ) . Remark 4.4.
While we will not use this in the sequel, an alternate way to viewthe A - A Hilbert bimodule structure on ˜ H is as follows. Given our basis β = { ξ n } Nn =1 ⊂ H , consider the probability space ( X, µ ) = Π n ( R , g ) where g is theGaussian measure on R . We can identify A = L ∞ ( X, µ ), and we denote by π n ∈ L ( X, µ ) the projection onto the n th copy of ( R , g ) so that the π n ’s are I.I.D.Gaussian random variables.We embed H into L ( X, µ ) linearly by the map η such that η ( ξ n ) = π n givenan orthogonal transformation T ∈ O ( H ), we associate to T the unique measure-preserving automorphism σ T ∈ Aut( A ) such that σ T ( η ( ξ )) = η ( T ξ ), for all ξ ∈ H .For each k we denote by A k = ( O n
7→ k δ β ( x ) k is a quantum Dirichlet form on L ( A ) (see [12, 43, 44]).In particular, it follows from [12] that D ( δ β ) ∩ A is a weakly dense ∗ -subalgebraand δ β | D ( δ β ) ∩ A is a derivation. Note that if we identify ˜ H with L k L ( A k ) as above then δ β can also be viewed asthe tensor product derivation δ β = N k δ k where δ k : L ( R , g ) → L ( R , g ) ⊗ L ( R , g )is the difference quotient derivation for each k , i.e. , δ k ( f )( x, y ) = f ( x ) − f ( y ) x − y . Lemma 4.5.
Using the above notation, δ β is a densely defined closed real deriva-tion, s ( H ) ⊂ D ( δ β ) , δ β ◦ s : H → ˜ H is an isometry, and for all T ∈ O ( H ) , σ T ( D ( δ β )) = D ( δ T β ) , and δ T β ( σ T ( a )) = ˜ T ( δ β ( a )) , for all a ∈ D ( δ β ) .Proof. The fact that s ( H ) ⊂ D ( δ β ), and that δ β ◦ s is an isometry follows from theformula δ β ( s ( ξ )) = ξ ⊗ Ω above.Moreover for ξ ∈ H we have δ T β ( σ T ( s ( ξ ))) = T ξ ⊗ Ω= ( T ⊗ T S )( ξ ⊗ Ω) = ˜
T δ β ( s ( ξ )) . By Lemma 4.3 this formula then extends to A , and since ˜ T acts on ˜ H unitarilyand A is a core for δ β we have that σ T ( D ( δ β )) = D ( δ T β ) and this formula remainsvalid for a ∈ D ( δ β ). (cid:3) Given an action of a countable discrete group Γ on a countable set S we mayconsider the generalized Bernoulli shift action of Γ on ( X, µ ) = Π s ∈ S ( R , g ) givenby γ ( r s ) s ∈ S = ( r γ − s ) s ∈ S . If we set H = ℓ S and consider the correspondingrepresentation π : Γ → U ( H ) then the generalized Bernoulli shift can be viewed asthe Gaussian action corresponding to π . Moreover we have that the canonical basis β = { δ s } s ∈ S is invariant to the representation, i.e. , π γ β = β , for all γ ∈ Γ.In this case by Lemma 4.5 we have that D ( δ β ) is σ γ invariant for all γ ∈ Γ and δ β ( σ γ ( a )) = ˜ π γ ( δ β ( a )), for all γ ∈ Γ, a ∈ D ( δ β ), where ˜ π : Γ → U ( ˜ H ) is the unitaryrepresentation given by ˜ π = π ⊗ π S . If we denote by N = A ⋊ Γ the correspondinggroup-measure space construction then using Lemma 4.3 we may define an N - N Hilbert bimodule structure on K = ˜ H ⊗ ℓ Γ which satisfies( au γ )( ξ ⊗ δ γ )( bu γ ) = ( a · β (˜ π γ ξ ) · β σ γ γ ( b )) ⊗ δγ γ γ , for all a, b ∈ A , γ , γ , γ ∈ Γ, and ξ ∈ ˜ H . We may then extend δ β to a closablederivation δ : ∗ -Alg( D ( δ β ) ∩ A, Γ) → K such that δ ( au γ ) = δ β ( a ) ⊗ u γ , for all a ∈ D ( δ β ), γ ∈ Γ.As above we denote by ζ α : N → N the unital, symmetric, c.p. resolvent mapsgiven by ζ α = ( α/ ( α + δ ∗ δ )) / , for α > M is a finite von Neumann algebra then we let Γ act on M triviallyand we may extend the derivation δ to ( A ⊗ M ) ⋊ Γ ∼ = ( A ⋊ Γ) ⊗ M by consideringthe tensor product derivation of δ with the trivial derivation (identically 0) on M .In this case the corresponding deformation of resolvent maps is just ζ α ⊗ id. Lemma 4.6.
Consider Ioana’s deformation α t on A corresponding to generalizedBernoulli shift as described above in Section 2.2. If M is a finite von Neumannalgebra and B ⊂ ( A ⊗ M ) ⋊ Γ is a subalgebra such that ζ α converges uniformly tothe identity on ( B ) as α → then α t converges uniformly to the identity on ( B ) as t → .Proof. The infinitesimal generator of Ioana’s deformation cannot be identified with δ as the α t ’s will converge uniformly on the algebra generated by s ( ξ ) for each ξ ∈ β , and ζ α will not have this property. However, it is not hard to check usingthe fact that both derivations arise as tensor product derivations that if ζ α are the N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 17 resolvent maps corresponding to the infinitesimal generator of α t then we have theinequality τ ( ζ α ( a ) a ∗ ) ≤ τ ( ζ α ( a ) a ∗ ), for all a ∈ A . Hence the lemma follows fromLemma 2.1 in [27] and Corollary 4.2 above. (cid:3) Remark 4.7.
It can be shown in fact that the deformation coming from the deriva-tion above, Ioana’s deformation, and the s-malleable deformation from the Gaussianaction, are successively weaker deformations. That is to say that one deformationconverging uniformly on a subset of the unit ball implies that the next deformationmust also converge uniformly.When we restrict the bimodule structure on K to the subalgebra L Γ we see thatthis is exactly the bimodule structure coming from the representation ˜ π = π ⊗ π S ,this give rise to the following lemma: Lemma 4.8.
Using the notation above, given
H < Γ we have the following: . LH K LH embeds into a direct sum of coarse bimodules if and only if π | H embedsinto a direct sum of left regular representations. . LH K LH weakly embeds into a direct sum of coarse bimodules if and only if π | H weakly embeds into a direct sum of left regular representations. . LH K LH has stable spectral gap if and only if π | H has stable spectral gap. . LH K LH is a compact correspondence if and only if π | H is a c -representation. . LH K LH is weakly mixing if and only if π | H is weakly mixing. L -rigidity and U fin -cocycle superrigidity In this section we use the tools developed above to prove U fin -cocycle superrigid-ity for the Bernoulli shift action which we view as the Gaussian action correspondingto the left-regular representation.To prove that a cocycle untwists we use the same general setup as Popa in [36].In particular, we use the fact that for a weakly-mixing action, in order to showthat a cocycle untwists it is enough to show that the corresponding s-malleable de-formation converges uniformly on the “twisted” subalgebra of the crossed productalgebra. The main difference in our approach is that to show that the s-malleabledeformation converges uniformly it is enough by Lemma 4.6 to show that the defor-mation coming from the Bernoulli shift derivation converges uniformly. This allowsus to use the techniques developed in [26], [27], [24], and [28] to analyze the cocycleon the level of the base space itself rather than the exponential of the space wherethe properties can be somewhat hidden. Theorem 5.1.
Let Γ be a countable discrete group. If L Γ is L -rigid then theBernoulli shift action with diffuse core of Γ is U fin -cocycle superrigid.Proof. Let
G ∈ U fin , then G ⊂ U ( M ) as a closed subgroup where M is a finiteseparable von Neumann algebra. Let c : Γ × X → G be a cocycle where X is theprobability space of the Gaussian action. Consider A = L ∞ ( X ), and ω : Γ →U ( A ⊗ M ) given by ω γ ( x ) = c ( γ, γ − x ) the corresponding unitary cocycle for theaction ˜ σ γ = σ γ ⊗ id. Note that ω γ γ = ω γ ˜ σ γ ( ω γ ), for all γ , γ ∈ Γ. Here weview a unitary element in A ⊗ M as map from X to U ( M ) (see [36] for a detailedexplanation).As noted above, the Bernoulli shift action with diffuse core is precisely theGaussian action corresponding to the left-regular representation; hence, by Lemma4.8 we have that as an L Γ- L Γ Hilbert bimodule K embeds into a direct sum of coarse correspondences. If we denote by f L Γ the von Neumann algebra gener-ated by { ˜ u γ } = { ω γ u γ } then the bimodule structure of f L Γ ( ∼ = L Γ) on K is thesame as the bimodule structure of L Γ on the correspondence coming from therepresentation γ Ad( ω γ ) ◦ ˜ π γ on ˜ H ⊗ L M . The A ⊗ M bimodule structureon ˜ H ⊗ L M = H ⊗ S ( H ′ ) ⊗ L M decomposes as a direct sum of bimodules H ⊗ S ( H ′ ) ⊗ L M = ⊕ ξ ∈ β S ( H ′ ) ⊗ L M where the bimodule structure on eachcopy of S ( H ′ ) ⊗ L M is given by Equation (1), and under this decomposition wehave Ad( ω γ ) ◦ ˜ π γ = π γ ⊗ (Ad( ω γ ) ◦ π S γ ). Therefore by Fell’s absorption principlethis representation is an infinite direct sum of left-regular representations; hence,we have that K also embeds into a direct sum of coarse correspondences when K isviewed as an f L Γ- f L Γ Hilbert bimodule.Since L Γ is L -rigid we have that the corresponding deformation ζ α convergesuniformly to the identity map on ( f L Γ) , by Lemma 4.6 we have that a correspondings-malleable deformation also converges uniformly to the identity on ( f L Γ) . Thus,by Theorem 3.2 in [36] the cocycle ω is cohomologous to a homomorphism. (cid:3) We end this paper with some examples of groups for which the hypothesis of theTheorem 5.1 is satisfied.It follows from [27] that if N is a nonamenable II factor which is non-prime,has property (Γ), or is w -rigid, then N is L -rigid. We include here another class of L -rigid finite von Neumann algebras, this class includes the group von Neumannalgebras of all generalized wreath product groups A ≀ X Γ where A is an infiniteabelian group and Γ does not have the Haagerup property, or Γ is a non-amenabledirect products of infinite groups. This is a special case of a more general resultwhich can be found in [28]. Theorem 5.2.
Let Γ be a countable discrete group which contains an infinite nor-mal abelian subgroup and either does not have the Haagerup property or containsan infinite subgroup Γ such that L Γ is L -rigid, then L Γ is L -rigid.Proof. We will use the same notation as in [27]. Suppose (
M, τ ) is a finite vonNeumann algebra with L Γ ⊂ M , and δ : M → L M ⊗ L M is a densely definedclosable real derivation.Since the maps η α converge point-wise to the identity we may take an appropriatesequence α n such that the map φ : Γ → R given by φ ( γ ) = Σ n − τ ( η α n ( u γ ) u ∗ γ )is well defined. If the deformation η α does not converge uniformly on any infinitesubset of Γ then the map φ is not bounded on any infinite subset and hence definesa proper, conditionally negative-definite function on Γ showing that Γ has theHaagerup property.Therefore if Γ does not have the Haagerup property then there must exist aninfinite set X ⊂ Γ on which the deformation η α converges uniformly. Similarly, ifΓ ⊂ Γ is an infinite subgroup such that L Γ is L -rigid then we have that thedeformation η α converges uniformly on the infinite set X = Γ .Let A ⊂ Γ be an infinite normal abelian subgroup. If there exists an a ∈ A such that a X = { xax − | x ∈ X } is infinite, then we have that the deformation η α converges uniformly on this set, and by applying the results in [27] it follows that η α converges uniformly on A ⊂ L ( A ). Since A is a subgroup in U ( LA ) which generates LA it then follows that η α converges uniformly on ( LA ) and hence also on ( L Γ) since A is normal in Γ. N COCYCLE SUPERRIGIDITY FOR GAUSSIAN ACTIONS 19 If a ∈ A and a X is finite then there exists an infinite sequence γ n ∈ X − X such that [ γ n , x ] = e , for each n . Thus if a X is finite for each a ∈ A then bytaking a diagonal subsequence we construct a new sequence γ n ∈ X − X such thatlim n →∞ [ γ n , a ] = e . Since η α also converges uniformly on X − X we may againapply the results in [27] to conclude that η α converges uniformly on A and henceon ( L Γ) as above. (cid:3) It has been pointed out to us by Adrian Ioana that in light of Corollary 1.3in [7] the above argument is sufficient to show that for a lattice Γ in a connectedLie group which does not have the Haagerup property, we must have that L Γ is L -rigid.We also show that L -rigidity is stable under orbit equivalence. The proof ofthis uses the diagonal embedding argument of Popa and Vaes [41]. Theorem 5.3.
Let Γ i y ( X i , µ i ) be free ergodic measure preserving actions for i = 1 , . If the two actions are orbit equivalent and L Γ is L -rigid then L Γ isalso L -rigid.Proof. Suppose L Γ ⊂ M and δ : M → H is a closable real derivation such that H as an L Γ bimodule embeds into a direct sum of coarse bimodules. Let N = L ∞ ( X , µ ) ⋊ L Γ = L ∞ ( X , µ ) ⋊ L Γ and consider the N ⊗ M bimodule ˜ H = L N ⊗H . If we embed N into N ⊗ M by the linear map α which satisfies α ( au γ ) = au γ ⊗ u γ for all a ∈ L ∞ ( X , µ ), and γ ∈ Γ , then when we consider the α ( N )- α ( N )bimodule ˜ H we see that this bimodule is contained in a direct sum of the bimodule L h α ( N ) , α ( L ∞ ( X , µ )) i coming from the basic construction of ( α ( L ∞ ( X, µ )) ⊂ α ( N )). Indeed, this follows because the completely positive maps corresponding toleft and right bounded vectors of the form 1 ⊗ ξ ∈ L N ⊗H are easily seen to livein L h α ( N ) , α ( L ∞ ( X , µ )) i .The α ( N )- α ( N ) bimodule L h α ( N ) , α ( L ∞ ( X , µ )) i is an orbit equivalence in-variant and is canonically isomorphic to the bimodule coming from the left regularrepresentation of Γ (see for example Section 1.1.4 in [32]). It therefore followsthat ˜ H when viewed as an α ( L Γ ) bimodule embeds into a direct sum of coarsebimodules.We consider the closable derivation 0 ⊗ δ : N ⊗ M → ˜ H as defined in Section 4.2and use the fact that L Γ is L -rigid to conclude that the corresponding deformationid ⊗ η α converges uniformly on the unit ball of α ( N ), (note that id ⊗ η α is the identityon α ( L ∞ ( X , µ )) = α ( L ∞ ( X , µ ))). In particular, id ⊗ η α converges uniformly on { α ( u γ ) | γ ∈ Γ } which shows that η α converges uniformly on { u γ | γ ∈ Γ } . As thisis a group which generates L Γ we may then use a standard averaging argumentto conclude that η α converges uniformly on the unit ball of L Γ , (see for exampleTheorem 4.1.7 in [30]). (cid:3) Remark 5.4.
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Jesse Peterson, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240
E-mail address : [email protected] Thomas Sinclair, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240
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