A RIGIDITY RESULT FOR ACTIONS OF

PROPERTY T GROUPS BY BERNOULLI SHIFTS

SORIN POPA

University of California, Los Angeles

Abstract. Let σ be the action of an infinite property T group G on the hyperfinite type II_{1} factor R = \overline{⊗_{g∈G}}(M_{2×2}(ℂ), tr)_{g}, by Bernoulli shifts. We prove that the cocycle actions obtained by reducing σ to the algebras pRp, for p non-trivial projections in R, cannot be perturbed to actions. We also prove that any 1-cocycle for σ vanishes. More generally, we calculate all 1-cocycles for actions of property T groups G by Bernoulli shifts of Connes-Størmer type and use this to provide an invariant that distinguishes these actions up to cocycle conjugacy.

0. Introduction.

The study of automorphisms in the theory of von Neumann algebras has often followed problems and phenomena already present in commutative ergodic theory. But starting with the early 70’s, with the advent of Tomita-Takesaki theory and Connes’ ground breaking work on the classification of factors, the full importance and the distinguishing features of this “non-commutative ergodic theory” have become increasingly evident.

A typically non-commutative aspect related to the study of automorphisms of a von Neumann algebra N is that such automorphisms are regarded both as elements in the automorphism group of N, AutN, and modulo perturbation by inner automorphisms, in the quotient group AutN/IntN. This leads to the study of two types of classification up to conjugacy for automorphisms, in AutN and respectively in AutN/IntN. It also leads to considering cocycles for the corresponding actions.

Along these lines, the more general problem is to study morphisms from groups G into AutN and into AutN/IntN, up to conjugacy. The former are called actions

Typeset by AMS-TEX

Supported in part by a NSF Grant 9801324.

of G on N. Not all faithful group morphisms σ from G into AutN/IntN can be lifted to actions. Thus, Nakamura and Takeda have already pointed out in ([NT]) that to any such morphism σ one can associate a scalar 3-cocycle μ ∈ H^{3}(G, 𝕋) of the group G, and that μ gives an obtruction for σ to be liftable to a genuine action. Also, it was shown in ([NT]) that the morphism σ has trivial obstruction μ if and only if it can be lifted to a cocycle action. It is a most interesting problem to decide whether any cocycle action of a group G can be perturbed by inner automorphisms to a genuine action of G. In other words, whether the 3-cocycle μ is the only obstruction for a σ to be liftable to an action.

One of the results in ([C3]) showed that this is indeed the case if G = ℤ/nℤ and N is an arbitrary type II_{1} factor. Then in ([Su]) it was proved that this is still the case for all finite groups G. It was further proved in ([Oc]) that the result actually holds true for all countable amenable groups G, but when N is equal to the hyperfinite type II_{1} factor R. Finally, the condition that N be hyperfinite was removed in ([Po1,9]). Thus, all cocycle actions of an amenable group G on an arbitrary type II_{1} factor N can be perturbed to genuine actions.

At the oposite end, Connes and Jones have found an example of a cocycle action of a property T group G ([K]) on the free group factor N = L(𝔽_{∞}) that cannot be perturbed to an action ([CJ]). It remained however as an open problem of whether such examples exist or not in the case N is the hyperfinite type II_{1} factor.

In this paper we solve this problem, by providing a class of examples of cocycle actions on the hyperfinite type II_{1} factor that cannot be perturbed to actions. More precisely, we prove that if σ is an action of an infinite disctere group G with the property T of Kazhdan on the hyperfinite type II_{1} factor R = \overline{⊗_{g∈G}}(M_{2×2}(ℂ), tr)_{g},

by Bernoulli shifts, and if p is an arbitrary projection in R, p ≠ 0, 1, then the cocycle actions obtained by reducing σ by p cannot be perturbed to genuine actions. The same result is proved for the group G = SL(2, ℤ) ⋊ ℤ^{2}, which doesn’t have the property T.

Our result also implies that given any two projections of different trace p, q in R the cocycle actions on pRp ≃ R ≃ qRq, obtained by reducing σ by p, respectively q, are not outer conjugate, i.e., they are not conjugate in AutR/IntR.

At the same time, we also prove a vanishing 1-cohomology result for the action σ, showing that any 1-cocycle (modulo scalars) w for σ is a coboundary, modulo the scalars. More precisely, if w : G → U(R) satisfies w_{g}σ_{g}(w_{h}) = w_{gh}, ∀g, h ∈ G, modulo ℂ, then there exists a unitary element v ∈ U(R) such that w_{g} = v^{∗}σ_{g}(v), ∀g ∈ G, modulo ℂ. Related to this, recall that by ([C2]) any unitary 1-cocycle for an arbitrary action of a finite group G on a type II_{1} factor vanishes. Our result shows that such a rigidity result may hold true for some special actions of infinite discrete groups as well.

This vanishing cohomology result for 1-cocycles also implies that if an action of G on R is cocycle conjugate to the Bernoulli shift σ, then it is actually conjugate to it.

More generally, we explicitly calculate all the 1-cocycles for actions of an infinite property T group G by Bernoulli shifts of Connes-Stormer type. These are actions σ_{λ} on the hyperfinite type II_{1} factor R obtained by restricting the Bernoulli shift of G on the hyperfinite type III_{λ} factor N = \overline{⊗_{g∈G}}(M_{2×2}(ℂ), φ_{λ})_{g} to its type II_{1} core.

We prove that all 1-cocycles for σ_{λ} are locally of the form vσ_{λ,g}(v^{∗}), with v ∈ N isometries normalizing the core R of N. As a consequence, for each 1-cocycle w the fixed point algebra of the action Adw ∘ σ follows atomic. Moreover, the scalar λ can be recuperated from the range of the trace on the set of minimal projections of these fixed point algebras, thus being an outer conjugacy invariant for σ. As a consequence, we obtain that the actions σ_{λ}, 0 < λ < 1, of the group G on R are mutually non outer conjugate.

Similarily, we consider arbitrary Connes-Størmer Bernoulli shifts σ of G on the core factor of N = \overline{⊗_{g∈G}}(M_{k×k}(ℂ), φ_{0})_{g} and show that the multiplicative subgroup

of ℝ_{+}^{∗} generated by {t_{i}/t_{j}}_{i,j}, where t_{i} > 0 are the diagonal elements that determine the faithful state φ_{0} of M_{k×k}(ℂ), is an outer conjugacy invariant for σ.

Note that if in the above construction the group G one starts with is ℤ, then the actions σ_{λ} are the usual Connes-Størmer Bernoulli shifts in ([CS]) and they have Connes-Størmer entropy equal to −t log t − (1 − t) log(1 − t), where t/1 − t = λ. Thus, the entropy does distinguish the actions σ_{λ}, 0 < λ < 1, up to conjugacy, but by ([C2]) these actions are all outer conjugate. Similarily for G an arbitrary amenable group, by ([Ka]) and respectively ([Oc]). Our result shows that in the case G has the property T, the Connes-Størmer Bernoulli shifts σ_{λ} become much more rigid, as they are not even outer conjugate. Also, rather than entropical, our invariant comes from the cohomology properties of the action.

We also obtain some results similar to the above for the free Bernoulli shifts of an infinite property T group G on the free group factor L(𝔽_{∞}) ≃ L(𝔽_{G}).

To prove these results we use the fact that if σ is a Bernoulli shift of an infinite property T group G on R then the identity automorphism on R\overline{⊗}R can be approximated by product-type automorphisms that are commuting with the action σ ⊗ σ on R\overline{⊗}R. Using this deformation of the identity and the property T of the group G, and assuming by contradiction that the reduced cocycle action by some projection p ∈ R can be perturbed to an action, one obtains non-zero intertwiners that implement the flip automorphism on appropriate elements of R\overline{⊗}R. When suitably interpreted, this forces p = 1 and the cocycle to be trivial.

The paper is organized as follows: In Section 1 we recall some basic definitions of actions, cocycle actions and 1-cocycles. In Section 2 we give some examples of actions, that are to be used in the paper, amply discussing the Bernoulli shifts and their generalized version considered by Connes and Stormer. In Section 3 we construct the cocycle actions obtained by reducing actions by projections (Proposition 3.1). Also, we introduce a class of non-trivial 1-cocycles for Connes-Størmer Bernoulli shifts that are needed in the sequel (Theorem 3.2).

In Section 4 we prove the main result of the paper, showing that the 1-cocycles constructed in Section 3 give the list of all 1-cocycles of the actions of property T groups by Connes-Størmer Bernoulli shifts (Theorem 4.1). In Section 5 we derive some consequences of the main theorem, proving the existence of cocycle actions that cannot be perturbed to actions (Corollary 5.7), the vanishing cohomology result for Bernoulli shifts of property T groups (Corollary 5.6) and showing that the outer conjugacy classes of the Connes-Størmer Bernoulli shifts can be distinguished by the “cohomology picture” of those actions, a structure that comes from the spectrum of the discrete decomposition of the type III_{λ} factors they are defined on (Theorem 5.3, 5.4). Also, we make several remarks. In Section 6 we prove the analogue result for free shifts. We include some remarks here as well.

I am very greatful to Dima Shlyakhtenko and Antony Wassermann for pointing out to me several pertinent references related to this work.

1. Preliminaries and notations.

Although the main results in this paper are stated in the framework of type II_{1} factors, their proofs will require some considerations on von Neumann factors of type III_{λ}, 0 < λ ≤ 1, as well. Thus, most of the definitions and notations on actions of groups and their cocycles that we recall in this section will be stated in the context of arbitrary von Neumann factors.

1.1. Actions and cross products. We denote by (N , φ) a pair consisting of a von Neumann factor N (typically of type II_{1} or of type III_{λ}, 0 < λ ≤ 1), with a normal faithful state φ on it (typically a trace or a generalized trace). The centralizer of φ is the set N_{φ} = {x ∈ N | φ(xy) = φ(yx), ∀x ∈ N }.

In case the von Neumann algebra N is specified to be a type II_{1} factor, we will use the notation N instead of N. In this case the state φ will always be taken to be the unique trace τ on N.

An automorphism of (N , φ) is an automorphism of N that preserves φ. We sometimes denote the group of all such automorphisms of N by Aut(N , φ). Note that any automorphism of (N , φ) normalizes N_{φ}.

An automorphism ρ of N is inner if there exists u in the unitary group of N, U(N), such that ρ(x) = Adu(x) = uxu^{∗}, ∀x ∈ N. Note that the set Int (N , φ) of inner automorphisms of N that preserve φ coincides with the set of automorphisms Adu with u ∈ U(N_{φ}). Int (N , φ) is clearly a normal subgroup of Aut (N , φ).

An automorphism ρ of the factor N is properly outer (or simply outer) if it is not inner.

Let G be a discrete group. An action of G on (N , φ) is a group morphism σ : G → Aut(N , φ). Note that in case N = N is a type II_{1} factor, any automorphism of N preserves the trace τ. We will simply denote by AutN the set of all automorphisms of N.

Recall that the cross product algebra associated to an action σ on (N , φ) denoted (N ⋊_{σ} G, φ), is the von Neumann algebra generated inside B(ℓ^{2}(G, L^{2}(N , φ))) by the unitaries u_{g} ∈ B(ℓ^{2}(G, L^{2}(N , φ))), g ∈ G, where

u_{g}(f)(h) = f(g^{−1}h), for f ∈ ℓ^{2}(G, L^{2}(N , φ)),

and by a copy of the algebra N given by

(b · f)(g) = σ_{g}^{−1}(b)f(g), for b ∈ N, f ∈ ℓ^{2}(G, L^{2}(N , φ)), g ∈ G.

together with the vector state φ(X) = ⟨Xξ_{φ}, ξ_{φ}⟩, where ξ_{φ} ∈ ℓ^{2}(G, L^{2}(N , φ)) is the function on G that takes the value ξ_{φ} at e and 0 elsewhere.

The cross product algebra N ⋊_{σ} G can alternatively be viewed as the completion (on bounded sequences) of the Hilbert algebra of finite formal sums Σ_{g∈G} u_{g}b_{g}, b_{g} ∈

N, with multiplication rules u_{g}u_{h} = u_{gh}, bu_{g} = u_{g}σ_{g}^{−1}(b), b = u_{e}b = 1b, for g, h ∈ G, b ∈ N, and ∗-operation (u_{g}b)^{∗} = u_{g−1}σ_{g}(b^{∗}), and with N -valued expectation

E (Σ_{g}u_{g}b_{g}) =^{def} b_{e} and scalar expectation

φ (∑ u_{g}b_{g}) =^{def} φ (E (∑ u_{g}b_{g})) = φ(b_{e}).

Recall that (N ⋊_{σ} G, φ) this way defined is itself a von Neumann algebra, with φ a faithful normal state. In case N is a finite von Neumann factor with φ a trace, then this cross product algebra is a finite von Neumann algebra itself, with φ = φ ∘ E a faithful trace on it.

If the action σ is properly outer, i.e., if σ_{g} is a properly outer automorphism, ∀g ≠ e, then N′∩N ⋊_{σ}G = ℂ. In particular, if σ is properly outer then M = N ⋊_{σ}G is a factor.

1.2. Cocycle actions. A cocycle action σ of G on (N , φ) is a map σ : G → Aut(N , φ) with the property that there exists a map v : G×G → U(N_{φ}) such that:

σ_{e} = id and σ_{g}σ_{h} = Ad v_{g,h}σ_{gh}, ∀g, h ∈ G (1.2.1) v_{g,h}v_{gh,k} = σ_{g}(v_{h,k})v_{g,hk}, ∀g, h, k ∈ G. (1.2.2)

A map v satisfying (1.2.2) is called a 2-cocycle for σ. The 2-cocycle is normalized if v_{g,e} = v_{e,g} = 1, ∀g ∈ G. Note that, since N is a factor, any 2-cocycle satisfies v_{e,e} ∈ ℂ. Thus any 2-cocycle v can be normalized by replacing it, if necessary, by v′_{g,h} = v_{e,e}^{∗} v_{g,h}, g, h ∈ G.

All 2-cocycles considered from now on will be normalized. Furthermore, all cocycle actions (in particular all actions) that we will consider in this paper are assumed to be properly outer, i.e., σ_{g} cannot be implemented by unitary elements in N , ∀g ≠ e.

Also, when given a cocycle action σ, we will sometimes specify from the beginning the 2-cocycle it comes with, thus considering it as a pair (σ, v).

Note that the 2-cocycle v is unique modulo perturbation by a scalar 2-cocycle μ. More precisely, v′ : G × G → U(N_{φ}), with v′_{e,e} = 1, satisfies conditions (1.2.1), (1.2.2) if and only if v′ = μv for some scalar valued function μ : G × G → 𝕋 satisfying μ_{e,e} = 1 and

(1.2.3)  μ_{g,h}μ_{gh,k} = μ_{h,k}μ_{g,hk}, ∀g, h, k ∈ G

A 2-cocycle v for the action σ is trivial (or it is a coboundary) if there exists a map w : G → U(N) such that w_{e} = 1 and v = ∂w, i.e.:

(1.2.3)  v_{g,h} = (∂w)_{g,h} =^{def} σ_{g}(w_{h}^{∗})w_{g}^{∗}w_{gh}, ∀g, h ∈ G.

The 2-cocycle v is weakly trivial if there exists w : G → U(N) such that w_{e} = 1 and v = ∂w modulo scalars, i.e.:

(1.2.4)  w_{g}σ_{g}(w_{h})v_{g,h}w_{gh}^{∗} ∈ ℂ1, ∀g, h ∈ G.

Note that this is equivalent to

(1.2.4′)  (Ad w_{g}σ_{g}) (Ad w_{h}σ_{h}) = Ad w_{gh}σ_{gh}, ∀g, h

i.e., to σ′_{g} =^{def} Ad w_{g}σ_{g} being an action.

A weakly trivial 2-cocycle is not necessarily trivial. In fact, if we take any scalar valued 2-cocycle, then the conditions (1.2.1), (1.2.2) and (1.2.4′) are satisfied for any genuine action σ of G (taking μ for v and w = 1). Thus μ is a weakly trivial 2-cocycle, in the above sense. But if we take N = ℂ1 it is not always true that given any μ like this there exists some w : G → 𝕋 such that μ = ∂w. In fact, for most groups G there do exist scalar 2-cocycles μ that are not trivial (or coboundary).

Let us also recall here some well known vanishing cohomology results. The first result along these lines is due to Connes, who considered the vanishing cohomology and classification problems for actions of the cyclic groups ℤ, ℤ/nℤ, n ≥ 2, on factors ([C2, 3]).

A general vanishing cohomology result for arbitrary finite groups was obtained by Sutherland ([Su] see also [J1]), who proved that any 2-cocycle from a cocycle action of a finite group G on an arbitrary type II_{1} factor N is coboundary.

In the same vein, Ocneanu proved that any 2-cocycle from a cocycle action of a countable amenable group G on the hyperfinite type II_{1} factor R is coboundary ([Oc]). It was then proved in ([Po1,9]) that the same vanishing cohomology result holds true for cocycle actions of amenable groups G on arbitrary type II_{1} factors N. Thus, in particular, any cocycle action of a countable amenable group on an arbitrary type II_{1} factor can be perturbed to a genuine action.

Connes and Jones provided the first example of a cocycle action of a countable discrete group G on a type II_{1} factor N that cannot be perturbed to an action ([CJ]). In their example, G had the property T of Kazhdan, with N being the free group factor L(𝔽_{∞}) (see the Appendix 2.1 at the end of this paper).

1.3. 1-cocycles for actions. Let us now take σ to be a genuine action of G on (N , φ). A map w : G → U(N_{φ}) satisfying condition

(1.3.1)  w_{g}σ_{g}(w_{h}) = w_{gh}, ∀g, h

is called a 1-cocycle for σ. Such a 1-cocycle for σ is a coboundary (or it is trivial) if there exists a unitary element w ∈ U(N) such that w_{g} = w^{∗}σ_{g}(w), ∀g. (Clearly, such maps w_{g} do satisfy the 1-cocycle condition (1.3.1)).

The map w is called a weak 1-cocycle if it satisfies the relation (1.3.1) modulo the scalars, i.e.,

(1.3.1’)  w_{g}σ_{g}(w_{h})w_{gh}^{∗} ∈ 𝕋1, ∀g, h ∈ G

Note that this is equivalent to Adw_{g} ∘ σ_{g} being an action. Note also that if w is a weak 1-cocycle then μ_{g,h} = w_{g}σ_{g}(w_{h})w_{gh}^{∗} is a scalar 2-cocycle.

A (weak) 1-cocycle w is weakly trivial (or weak cobouboundary) if there exists a unitary element w ∈ U(N) such that ww_{g}σ_{g}(w)^{∗} ∈ 𝕋1, ∀g.

Two (weak) 1-cocycles w, w′ of the action σ are equivalent if there exists a unitary element v ∈ N such that w′_{g} = vw_{g}σ_{g}(v)^{∗}, ∀g ∈ G (resp. modulo scalars). Thus, a weak 1-cocycle is weakly trivial iff it is equivalent to a scalar valued weak 1-cocycle (N.B.: this are just plain scalar functions on G). Note that the scalar valued genuine 1-cocycles are just characters of G.

Recall that by Connes 2 by 2 matrix trick ([C3]) any 1-cocycle for an action of a finite group G on an arbitrary type II_{1} factor is trivial (the result in [C3] is for G = ℤ/nℤ, but the proof there goes the same way for G finite; for all the results on finite groups see also [J1]).

1.4. Generalized 1-cocycles. Like in 1.3, let σ be a genuine action of the discrete group G on the factor N. We consider the following general version of 1-cocycles: Let p be a non-zero projection in N_{φ} and w : G → N_{φ} be so that w_{g} ∈ N_{φ} are partial isometries satisfying w_{g}w_{g}^{∗} = p, w_{g}^{∗}w_{g} = σ_{g}(p), g ∈ G, with w_{e} = p. If w satisfies the condition

(1.4.1)  w_{g}σ_{g}(w_{h}) = w_{gh}, ∀g, h ∈ G

then w is called a generalized 1-cocycle for σ. If w satisfies the weaker condition

(1.4.1’)  w_{g}σ_{g}(w_{h})w_{gh}^{∗} ∈ 𝕋p, ∀g, h ∈ G

then it is called a generalized weak 1-cocycle. Note that (1.4.1’) is equivalent to σ′_{g} = Ad(w_{g})σ_{g|pN p} being an action of G on pN p. The projection p is called the support of w. Also, like for weak 1-cocycles, note that if w satisfies (1.4.1’) then the scalar valued function μ_{g,h} satisfying w_{g}σ_{g}(w_{h}) = μ_{g,h}w_{gh} is a 2-cocycle.

Note that in case its support is equal to 1, a generalized (weak) 1-cocycle w is a (weak) cocycle.

The generalized 1-cocycle w is trivial if there exists a partial isometry v ∈ N such that vv^{∗} = σ_{g}(vv^{∗}) and w_{g} = v^{∗}σ_{g}(v), for all g ∈ G. Similarily, if w is a weak generalized cocycle and the above condition is satisfied modulo scalars then the w is weakly trivial.

Note that if σ is ergodic then it cannot have generalized 1-cocycles of support ≠ 1. Also, if w is a generalized (weak) 1-cocycle of support p and q ∈ pNp is a projection in the fixed point algebra of σ′ = Adw ∘ σ then {qw_{g}}_{g} is a generalized (weak) 1-cocycle for σ as well.

1.5. Cocycle conjugacy of actions. Two cocycle actions (σ, v), (σ′, v′) of G on (N , φ) are cocycle conjugate if there exists an automorphism ρ of (N , φ) such that the following conditions are satisfied:

(1.5.1)  ρσ_{g} ρ^{−1} = Ad w_{g}σ′_{g}, ∀g.

(1.5.2)  ρ(v_{g,h}) = w_{g}σ′_{g}(w_{h})v′_{g,h} w_{gh}^{∗}, ∀g, h.

The cocycle actions σ, σ′ are outer conjugate (or weakly cocycle conjugate) if condition (1.5.1) is satisfied. Note that outer conjugacy is equivalent to the image morphisms of σ, σ′ in Aut (N , φ)/Int (N , φ) being conjugate in Aut (N , φ)/Int (N , φ) by an automorphism of (N , φ).

The (cocycle) actions σ, σ′ are conjugate if there exists an automorphism ρ of (N , φ) such that conditions (1.5.1) is satisfied with w = 1. We then write σ ∼ σ′.

Recall that Connes proved in ([C3]) that in the case N is isomorphic to the hyperfinite type II_{1} factor R, any two actions of ℤ/nℤ on it are conjugate. Then Jones proved that any two actions of an arbitrary finite group G on R are conjugate. Also, it was proved in ([C2]) that any two actions of ℤ on R are cocycle conjugate. Finally, Ocneanu proved in ([Oc]) that any two actions of an arbitrary amenable group G on R are cocycle conjugate.

2. Examples of actions.

Let us mention some interesting classes of examples of actions and cocycle actions, that will intervene in the sequel.

2.1. Bernoulli shifts. Let (N_{0}, τ_{0}) be a finite von Neumann factor with a faithful normal trace τ_{0}, τ_{0}(1) = 1. Let G be a discrete group and (N, τ) =^{def} \overline{⊗_{g∈G}}(N_{g}, τ_{g}), where (N_{g}, τ_{g}) = (N_{0}, τ_{0}), ∀g ∈ G. We let σ : G → Aut N be defined

by σ_{g} ( ⊗_{h∈G} b_{h}) = ⊗_{h∈G} b′_{h}, where ⊗_{h∈G} b_{h} ∈ \overline{⊗_{h∈G}} N_{h} has only finitely many entries

≠ 1 and b′_{h} = b_{g^{−1}h}.

An action σ defined this way is called a Bernoulli shift. When we need to be more specific, we’ll say that σ is a (N_{0}, τ_{0})-Bernoulli shift. Note that, with this terminology, we have that σ ⊗ σ is a N_{0}\overline{⊗}N_{0}-Bernoulli shift.

It is well known that if either G is infinite and N_{0} ≠ ℂ, or if G is finite and N_{0} has no atoms, then the Bernoulli shift σ is a properly outer action of G on N. Moreover, if G is infinite then σ is also ergodic (it is even strongly mixing: see 2.4 hereafter).

Also, note that if N_{0} is a factor (respectively an approximately finite dimensional finite von Neumann algebra) then so is N.

In particular, by taking N_{0} to be isomorphic to the hyperfinite type II_{1} factor, this shows that any discrete countable group G admits a properly outer (+ ergodic,

if |G| = ∞) action on R (≃ \overline{⊗_{g∈G}} R_{g}).

2.2. Connes-Størmer Bernoulli shifts. Let us now consider a generalized version of the above example, as introduced in ([CS]). To this end, we first need to recall some basic facts about ITPF1 factors and, more generally, about factors having a discrete decomposition ([AW, C4,5]).

2.2.0. Factors with discrete decomposition. We let (N , φ) be a von Neumann factor with the centralizer N = N_{φ} satisfying N′ ∩ N = ℂ. We denote

V = {v ∈ N | v^{∗}v = 1, vNv^{∗} = vv^{∗}Nvv^{∗}, φ(vX) = φ(vv^{∗})φ(Xv), ∀X ∈ N }.

Remark that in fact for an isometry v ∈ N to be in V it is sufficient to satisfy φ(vX) = φ(vv^{∗})φ(Xv), ∀X ∈ N (see e.g. [C4,5]).

We denote by H_{1} = {φ(vv^{∗} | v ∈ V} and for each β ∈ H_{1}, denote V_{β} = {v ∈ V | φ(vv^{∗}) = β}. Thus, V = ∪_{β}V_{β}. Also, we clearly have V_{β}V_{γ} = V_{βγ} and V_{1} = U(N). Actually, for any two isometries v, v′ ∈ V_{β} the element w = v′v^{∗} is a partial isometry in N with v′ = wv. Also, it is easy to see that if v ∈ V_{β}, v′ ∈ V_{β′}, with β ≠ β′, then E(v^{∗}v′) = 0 = E(v′v^{∗}), where E = E_{N}^{φ} is the φ-preserving conditional expectation of N onto N.

We further assume that the set V generates N as a von Neumann algebra. By ([C5]), this is equivalent to (N , φ) having a discrete decomposition over N. More precisely, if for each β ∈ H_{1} we choose an isometry v_{β} ∈ V_{β} then each X ∈ N has a cross product type decomposition X = Σ_{β}E(Xv_{β}^{∗})v_{β} + Σ_{β}v_{β}^{∗}E(v_{β}X).

In fact, if we denote by H the multiplicative group generated by H_{1}, i.e., H = H_{1} ∪ H_{1}^{−1}, then (N\overline{⊗}B(ℓ^{2}(ℕ)) ⊂ N \overline{⊗}B(ℓ^{2}(ℕ)) is isomorphic to (N\overline{⊗}B(ℓ^{2}(ℕ)) ⊂ (N\overline{⊗}B(ℓ^{2}(ℕ)) ⋊ H). Moreover, if we denote by p_{0} ∈ N\overline{⊗}B(ℓ^{2}(ℕ)) a projection of the form 1_{N} ⊗ q_{0}, where q_{0} is a one dimensional projection, and by {u_{β}}_{β∈H} the canonical unitaries in N \overline{⊗}B(ℓ^{2}(ℕ)) ≃ N\overline{⊗}B(ℓ^{2}(ℕ)) ⋊ H that implement the action of the group H then p_{0}u_{β}p_{0} = v_{β} and p_{0}u_{β^{−1}}p_{0} = v_{β}^{∗}, ∀β ∈ H_{1}.

In terms of modular theory, it is easy to see that (N , φ) has a discrete decomposition as above iff the modular automorphism group associated with φ is almost periodic and has pure point spectrum. We call the group H the spectrum of the discrete decomposition.

Examples of such situations are, for instance, when (N , φ) is a type II_{1} factor N with its trace, because then V equals the set of unitary elements in N, with H = {1}. Another example is when (N , φ) is a type III_{λ} factor, 0 < λ < 1, with φ its generalized trace (cf. [C4, 5]). In this case H = {λ^{n} | n ∈ ℤ} and H_{1} = {λ^{n} | n ≥ 0}. The type III_{1} factors that can be obtained as tensor products of some type III_{λ_{i}} factors, 0 < λ_{i} < 1, i = 1, 2, ..., k, have discrete decompositions as well. In this case H is the multilicative subgroup of ℝ_{+}^{∗} generated by {λ_{i}}_{i}.

2.2.1. ITPF1 factors. An important class of concrete factors having a discrete decomposition is as follows: Let (N_{0}, φ_{0}) to be a finite dimensional factor with a faithful state on it. Let G be an infinite discrete group as before and (N , φ) =^{def} \overline{⊗_{g∈G}}(N_{g}, φ_{g}), where (N_{g}, φ_{g}) = (N_{0}, φ_{0}), ∀g ∈ G, be the corresponding infinite

tensor product, or ITPF1 factors ([P, AW]).

In case N_{0} = M_{2×2}(ℂ) and φ_{0} is given by the diagonal element (t, 1 − t) with t < 1/2, then N is a type III_{λ} factor, where λ = t/1 − t, and φ is its generalized trace. Thus, if N = N_{φ} then (N , φ) has a discrete decomposition over N.

In general, if N_{0} = M_{k×k}(ℂ) and the state φ_{0} is given by a diagonal element with spectrum 0 < t_{1} ≤ t_{2}... ≤ t_{k} then N is a type II_{1} factor if t_{1} = ... = t_{k}, it is a type III_{λ} factor, for some 0 < λ < 1, if the multiplicative group generated by {t_{i}/t_{j}}_{i,j} in ℝ_{+}^{∗} is equal to {λ^{n} | n ∈ ℤ} and it is a type III_{1} factor if the the multiplicative group generated by {t_{i}/t_{j}}_{i,j} is not single generated. (Thus, generically, the IPTF1 factors N are of type III_{1}.)

In this general case the centralizer N = N_{φ} of the state φ still satisfies N′∩N = ℂ and in fact (N , φ) has a disrete decomposition over N, with spectrum H ⊂ ℝ_{+}^{∗} given by the multiplicative group generated by {t_{i}/t_{j} | 1 ≤ i, j ≤ k} ⊂ ℝ_{+}^{∗}.

2.2.2. Generalized Bernoulli shifts. With N = \overline{⊗}_{g}(M_{k×k}(ℂ), φ_{0})_{g} as above, let σ : G → AutN be defined as follows: first define σ to be the Bernoulli shift on the algebraic infinite tensor product ⊗_{g}N_{g}, as before; then note that on this dense sbalgebra of N we have φ ∘ σ_{g} = φ, ∀g; thus each σ_{g} can be extended to a φ-preserving automorphism σ_{g} on all N. Note that when G is infinite σ is properly outer and ergodic ([CS]). Such an action is called a generalized Bernoulli shift or a (N_{0}, φ_{0})-Bernoulli shift.

Note that the generalized Bernoulli shift σ leaves each one of the sets of isomteries V_{β} invariant, i.e., σ_{g}(V_{β}) = V_{β}, ∀g ∈ G, ∀β ∈ H_{1}. In fact, if v ∈ V then w_{g} = vσ_{g}(v^{∗}) are partial isometries in N with σ_{g}(v) = w_{g}^{∗}v.

2.2.3 Connes-Størmer Bernoulli shifts. Let (N , φ), σ, N = N_{φ} ⊂ N be as before. Since φ is invariant to σ, σ_{g}(N) = N, ∀g. Moreover, by ([AW, C1,4]), the von Neumann algebra N (which satisfies N′ ∩ N = ℂ) is isomorphic to the hyperfinite type II_{1} factor R.

Moreover, the action σ_{g} = σ_{g|N} is properly outer and ergodic on N ([CS]). Such actions are called Connes-Størmer Bernoulli shifts of G on the hyperfinite type II_{1} factor.

Note that if σ is a (N_{0}, φ_{0})-Bernoulli shift on (N , φ) as before then σ ⊗ σ is a (N_{0} ⊗ N_{0}, φ_{0} ⊗ φ_{0})-Bernoulli shift on on (N \overline{⊗}N , φ ⊗ φ). However, if we let Ñ be the centralizer of φ ⊗ φ in N \overline{⊗}N and denote by θ_{g} the restriction of σ_{g} ⊗ σ_{g} to Ñ, then θ is a Connes-Størmer Bernoulli shift, with Ñ ⊃ N\overline{⊗}N and the restriction of θ to N\overline{⊗}N equals σ_{|N} ⊗ σ_{|N}, but in general Ñ is different from N\overline{⊗}N and afortiori θ ≠ σ ⊗ σ as well.

2.3. Free Bernoulli shifts. Let (B_{0}, τ_{0}) be a finite von Neumann algebra with a finite faithful normal trace on it. Let {(B_{g}, τ_{g})}_{g∈G} be copies of (B_{0}, τ_{0}) indexed by the discrete group G. Define (N, τ) =^{def} ∗_{g∈G}(N_{g}, τ_{g}) to be the free product of

the algebras (B_{g}, τ_{g}) (see [V1]). Then we define σ_{g} (∗_{h}b_{h}) = ∗_{h}b′_{h}, where b′_{h} = b_{g^{−1}h}

(see e.g. [S, Po3]). Such an action is called a free Bernoulli shift of the group G.

Note that by ([Dy1]), if G is infinite and B_{0} ≠ ℂ then N is a factor. Moreover, σ is then easily seen to be properly outer and ergodic.

Also, by ([Dy1]), if B_{0} is AFD, or if B_{0} is a free group factor, then N is isomorphic to the free group factor L(𝔽_{∞}).

Thus, any discrete countable group G acts properly outer (+ ergodic, if |G| = ∞) on the free group factor L(𝔽_{∞}).

Note that one can also define a Connes-Stormer version of the free Bernoulli shifts, by using Dykema’s free version of the IPTF1 factors ([Dy2]). Thus, one starts with a fixed finite dimensional von Neumann algebra with a faithful state on it, (B_{0}, φ_{0}). One defines (N , φ) =^{def} ∗_{g∈G}(B_{g}, φ_{g}), where (B_{g}, φ_{g}) = (B_{0}, φ_{0}), ∀g ∈ G,

noting that φ is quasi-periodic on N ([Sh]), being a generalized trace if B_{0} = M_{2×2}(ℂ). One takes N to be the centralizer of the state φ. Then one considers the free Bernoulli shift σ on N (which we could call a generalized free Bernoulli shift), noting that it preserves φ. Finally, one takes the free Connes-Stormer Bernoulli shift to be the restriction of σ to N, which by ([Dy2]) is isomorphic to L(𝔽_{∞}) (see also [Sh]).

2.4. Bernoulli shifts are mixing. Both the “hyperfinite” (2.1, 2.2) and the “free” (2.3) Bernoulli shifts have the property that they are “very ergodic”. More precisely, they satisfy the following property.

2.4.1. Definition. Let N be a von Neumann algebra with a faithful normal state φ. Let σ : G → AutN be a properly outer action of the discrete group G on N that preserves φ. The action σ is strongly mixing if

lim_{g→∞}φ(y_{1}σ_{g}(x)y_{2}) = φ(x)φ(y_{1}y_{2}), ∀x, y_{1}, y_{2} ∈ N .

Strongly mixing actions are clearly ergodic. In fact their ergodicity is so “strong” that it goes through to tensor products. More precisely, one has:

2.4.2. Lemma. Let σ_{1} be a strongly mixing action of a group G on the von Neumann algebra (N , φ) and σ_{0} an arbitrary action of G on a von Neumann algebra (N_{0}, φ_{0}). Denote by θ′ the action of G on N \overline{⊗}N_{0} given by θ′_{g} = σ_{g} ⊗ σ_{0,g}, g ∈ G. Then (N \overline{⊗}N_{0})^{θ′} = ℂ ⊗ N_{0}^{σ_{0}}.

Proof. Let φ′ = φ ⊗ φ_{0}. Let x ∈ (N \overline{⊗}N_{0})^{θ′}, x ≠ 0 and ε > 0. Denote by Y_{0} the φ′-preserving expectation of x onto ℂ ⊗ N_{0} and X_{0} = x − Y_{0}. Note that Y_{0} ∈ ℂ ⊗ N_{0}^{σ_{0}} ⊂ (N \overline{⊗}N_{0})^{θ′}. Thus X_{0} belongs to (N \overline{⊗}N_{0})^{θ′} as well.

By the density of the algebra N ⊗ N_{0} in N \overline{⊗}N_{0}, there exists an orthonormal system x_{0} = 1, x_{1}, x_{2}, ..., x_{n} ∈ N and elements y_{0} = Y_{0}, y_{1}, y_{2}, ..., y_{n} ∈ N_{0} such that

if we denote x′ = Σ_{i=0}^{n}x_{i} ⊗ y_{i} then we have

∥x − x′∥_{φ′} < ε/(3∥x∥_{φ′}) and∥x′∥_{φ′} ≤ ∥x∥_{φ′}.

Since σ is mixing, there exists g ∈ G such that

Σ_{i,j=1}^{n}|φ(σ_{g}(x_{i})x_{j}^{∗})| |φ_{0}(σ_{0,g}(y_{i})y_{j}^{∗})| < ε/3.

Thus, if we denote X′_{0} = Σ_{i=1}^{n}x_{i} ⊗ y_{i} = x′ − Y_{0} then we have

|φ′(θ′_{g}(X′_{0})X′_{0}^{∗})| < ε/3 and∥X_{0} − X′_{0}∥_{φ′} = ∥x_{0} − x′_{0}∥_{φ′} < ε/(3∥x∥_{φ′}).

As a consequence we get

∥X_{0}∥_{φ′}^{2} = φ′(θ′_{g}(X_{0})X_{0}^{∗})
≤ |φ′(θ′_{g}(X′_{0})X′_{0}^{∗})| + 2∥X_{0} − X′_{0}∥_{φ′}∥X_{0}∥_{φ′} < ε.

Since ε > 0 was arbitrary, it follows that X_{0} = 0 so that x = Y_{0} belongs to ℂ⊗N_{0}^{σ_{0}}. Q.E.D.

2.4.3. Corollary. If σ is a strongly mixing action of a group G on the von Neumann algebra (N , φ) then ℂ1 ⊂ N is the only finite dimensional subspace of L^{2}(N , φ) which is invariant to σ_{g}, ∀g ∈ G.

Proof. By applying 2.4.2 to σ_{0} = σ and by taking into account that the Hilbert space of Hilbert-Schmidt operators on L^{2}(N , φ), with the action implemented by σ on it, can be naturally identified with L^{2}(N , φ)\overline{⊗}L^{2}(N , φ), with the action σ ⊗ σ on it, it follows that the one dimensional projection of L^{2}(N , φ) onto ℂ1 is the only fixed point for the former. Q.E.D.

As mentioned before, let us point out that the hyperfinite and the free Bernoulli shifts provide examples of strongly mixing actions.

2.4.4. Lemma. All hyperfinite Bernoulli shifts (including the “generalized” and the “Connes-Størmer” ones) and the free Bernoulli shifts are strongly mixing.

Proof. Let (N_{0}, φ_{0}) be either the hyperfinite type II_{1} factor with φ_{0} = τ its trace, or a finite dimensional factor with φ_{0} a faithful state on it. Let G be a discrete

group and (N , φ) =^{def} \overline{⊗_{g∈G}}(N_{g}, φ_{g}), where (N_{g}, φ_{g}) = (N_{0}, φ_{0}), ∀g ∈ G.

By using first the Kaplansky density theorem then the Cauchy-Schwartz inequality, it follows that it is suficient to prove the statement for y_{1,2} in the algebraic tensor product ⊗_{g}(N_{g}, φ_{g}).

Assume that in this infinite algebraic tensor product y_{1,2} are supported by a finite subset S ⊂ G. Since each such element is a linear combination of elements of the form ⊗_{g∈S}e_{i_{g},j_{g}}^{g}, where e_{k,l}^{g} ∈ N_{g} ≃ N_{0} are matrix units chosen so that φ_{g} is

supported on the diagonal {e_{kk}^{g}}_{k}.

It follows that it is suficient to prove the statement for y_{1,2} of the form ⊗_{g∈S}e_{i_{g},j_{g}}^{g}.

But for each such element y_{i} there exists a non-zero scalar c_{i} such that φ(x′y_{i}) = c_{i}φ(y_{i}x′), ∀x′ ∈ N.

Thus φ(y_{1}σ_{g}(x)y_{2}) = c_{2}φ(y_{2}y_{1}σ_{g}(x)). By approximating x with elements in the algebraic tensor product ⊗_{g}(N_{g}, φ_{g}) and by applying again Cauchy-Schwartz inequality, it follows that in calculating the limit lim_{n→∞}φ(y_{2}y_{1}σ_{g}(x)), it is sufficient to take x ∈ ⊗_{g}(N_{g}, φ_{g}). But for such x, y_{1}, y_{2} we clearly have lim_{n→∞}φ(y_{2}y_{1}σ_{g}(x)) =

φ(y_{2}y_{1})φ(x) = c_{2}^{−1}φ(y_{1}y_{2})φ(x). Altogether, it follows that lim_{n→∞}φ(y_{1}σ_{g}(x)y_{2}) = φ(y_{1}y_{2})φ(x).

The proof of the free group case is similar, so we leave it as an exercise for the reader (see e.g. [S]). Q.E.D.

2.5. Product type actions. Let us also mention a class of examples of properly outer actions that are not ergodic. Thus, with the notations of the examples 2.1-2.2, let G_{0} ⊂ AutN_{0} be a subgroup of automorphisms with α_{0,v} ≠ id, ∀v ≠ 1, and φ_{0} ∘ α_{0,v} = φ_{0}, ∀v ∈ G_{0}, where α_{0,v} is the image of v ∈ G_{0} in AutN_{0}. G_{0} will be considered with the topology inherited from AutN_{0}.

Let N denote the von Neumann algebra constructed from (N_{0}, φ_{0}) by infinite tensor product indexed by the infinite discrete group G (or simply indexed by ℕ),

as in 2.2. We let α : G_{0} → Aut N be defined by α_{v} ( ⊗_{h∈G} b_{h}) = ⊗_{h∈G} b′_{h}, where

b′_{h} = α_{0,v}(b_{h}), ∀h ∈ G.

It is easy to see that α is a continuous, properly outer action of G_{0} on (N , φ). In particular, since α preserves φ, it leaves the centralizer N = N_{φ} of φ invariant, thus implementing on it an action. Moreover, if N_{0} is finite dimensional then the action α has fixed point algebra N^{α} satisfying N^{α}′ ∩ N = ℂ1 ([W]). Thus, α is far from being ergodic.

Note that the action α commutes with the (N_{0}, φ_{0})-Bernoulli shift σ on (N , φ) considered in 2.1, 2.2. Thus, α_{|N} commutes with σ_{|N} as well.

A particular case of product type automorphism is the flip authomorphism on N \overline{⊗}N, that takes x ⊗ y into y ⊗ x. This automorphism will play an important role in Section 3. In fact, we will need the existence of some continuous path of automorphisms that relate the identity to the flip automorphism.

2.5.1. Lemma. With the notations used in 2.2, let (N_{0}, φ_{0}) be a finite dimensional factor and σ be the (N_{0}, φ_{0})- Bernoulli shift of the group G on the factor (N , φ). Let Ñ be the centralizer of φ ⊗ φ on N \overline{⊗}N and θ be the Connes-Stormer Bernoulli shift on Ñ obtained by restricting the (N_{0} ⊗ N_{0}, φ_{0} ⊗ φ_{0})-Bernoulli shift σ ⊗ σ from N \overline{⊗}N to Ñ. Then there exists a continuous product type action α of ℝ on (N \overline{⊗}N , φ ⊗ φ) such that:

1^{∘}. α is φ ⊗ φ-preserving and leaves Ñ invariant, thus implementing on it an action still denoted by α.

2^{∘}. α commutes with the generalized Bernoulli shift σ ⊗ σ and its restriction to Ñ commutes with θ.

3^{∘}. α_{±1}(N ⊗ ℂ) = ℂ ⊗ N.

If in addition we assume (N_{0}, φ_{0}) = (M_{2^{n}×2^{n}}(ℂ), φ_{0}) ≃ ⊗_{1≤i≤n}(M_{2×2}(ℂ), φ_{0,i})_{i} ,

for some n ≥ 1, then α can be constructed so that there exists a period 2 product type automorphism β on N \overline{⊗}N that preserves φ ⊗ φ, leaves Ñ invariant and satisfies the conditions:

4^{∘}. N ⊗ ℂ1 ⊂ Ñ^{β}.

5^{∘}. βα_{t}β^{−1} = α_{−t}, ∀t.

Proof. Let us first assume σ is of step 2. We use the notations of 2.2.

We let {e_{ij}}_{i,j=1,2} be a matrix unit for N_{0} = M_{2×2}(ℂ) such that the state φ_{0} on N_{0} is given by a dyagonal operator Σ_{i}λ_{i}e_{ii}.

We first take the action α_{0} of ℝ on N_{0} ⊗N_{0} to be defined by α_{0,t}(x) = Ad(v_{t})(x), t ∈ ℝ, with the unitary v_{t} ∈ N_{0} ⊗ N_{0} being given by

1  0  0  0
0 cos πt/2 sin πt/2 0
0 − sin πt/2 cos πt/2 0
0  0  0  1

Then we define the period 2-automorphism β_{0} on N_{0} ⊗ N_{0} as Adu_{0}, where the unitary u_{0} is given by:

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )

Note that the unitaries v_{t}, t ∈ ℝ, and u_{0} belong to the centralizer of the state φ_{0} ⊗ φ_{0} on N_{0} ⊗ N_{0}. Moreover, u_{0} lies in ℂ ⊗ N_{0}. An easy calculation shows that β_{0}α_{0,t}β_{0}^{−1} = α_{0,−t} and that Ad(v_{±1})(N_{0} ⊗ ℂ) = ℂ ⊗ N_{0}.

Define α_{t} to be the product-type action ⊗_{g}(α_{0,t})_{g} and β to be the automorphism ⊗_{g}(β_{0})_{g}. Then both α and β leave the centralizer Ñ of the product state φ ⊗ φ = ⊗_{g}(φ_{0} ⊗ φ_{0})_{g} invariant.

Moreover, by the way α, β are defined, conditions 1^{∘} − 5^{∘} are trivially satisfied. Note that by taking tensor products of the above contsruction, this also proves

the case (N_{0}, φ_{0}) = ⊗_{1≤i≤n}((M_{2×2}(ℂ), φ_{0,i})_{i}.

Now, in the general case when N_{0} = M_{k×k}(ℂ), let {e_{ij}}_{i,j} be a matrix unit with φ_{0} given by a dyagonal operator in Alg{e_{ii}}_{i}, as before. Let also

v_{t} = Σ_{i}e_{ii} ⊗ e_{ii}

+ Σ_{i<j} cos πt/2(e_{ii} ⊗ e_{jj} + e_{jj} ⊗ e_{ii}) + sin πt/2(e_{ij} ⊗ e_{ji} − e_{ji} ⊗ e_{ij}).

Then clearly 1^{∘} − 3^{∘} are satisfied. Q.E.D.

3. Examples of cocycles.

We first explain a method for producing a family of cocycle actions from a given action. Then we construct a class of examples of non-trvial 1-cocycles for actions on type II_{1} factors that come from a discrete decomposition.

Thus, we first let (N , φ) be a von Neumann factor with centralizer N = N_{φ}. We assume N to be a factor itself. We consider properly outer cocycle actions (σ, v) of the discrete group G on (N , φ). Thus, N is invariant to any such σ.

Let p ∈ P(N) be a non-zero projection. Since N is a factor and σ_{g} are tace preserving on N, for each g ∈ G the projections p and σ_{g}(p) are equivalent in N. Choose partial isometries w_{g} ∈ N_{φ} of left support p such that w_{g}σ_{g}(p)w_{g}^{∗} = p and w_{e} = p.

3.1. Proposition. Let σ^{p} : G → Aut(N_{p}) be defined by σ_{g}^{p}(pxp) = w_{g}σ_{g}(pxp)w_{g}^{∗}, for x ∈ N. Let also v_{g,h}^{p} = w_{g}σ_{g}(w_{h})v_{g,h}w_{gh}^{∗}, g, h ∈ G, which we regard as an element in (N_{φ})_{p} = pN_{φ}p. Then (σ^{p}, v^{p}) is a properly outer cocycle action of G on (N_{p}, φ_{p}, where φ_{p}(x) = φ(pxp)/φ(p), x ∈ pN p. Moreover, up to cocycle conjugacy, the cocycle action (σ^{p}, v^{p}) does not depend on the choice of the partial isometries w_{g}.

Proof. A straightforward calculation shows that v^{p} checks the conditions (1.2.1), (1.2.2) with respect to σ^{p}. Q.E.D.

Assume now N is the type II_{1} factor N. Since (pNp, σ^{p}) ≃ (qNq, σ^{q}) if p, q ∈ P(N) have same trace, from Lemma 3.1 we see that the isomorphism class of (N_{p}, σ^{p}), up to cocycle conjugacy, only depends on τ(p). Thus, we can denote it by (N_{t}, σ^{t}), for each 0 < t ≤ 1. More generally. we consider the following:

3.1.1. Notation. If t > 0 then we denote by (N_{t}, σ^{t}) the reduced of (M_{n×n}(N), id ⊗ σ), for some n ≥ t, by a projection p with normalized trace τ(p) satisfying nτ(p) = t. Again, by the above Lemma, up to cocycle conjugacy this action only depends on t. We call (N_{t}, σ^{t}) the amplification of (N, σ) by t.

Note that if in 3.1 we take σ to be a genuine action (so v = 1) and take the projection p in the fixed point algebra N^{σ} then the 2-cocycle v^{p} is vanishing, in other words the cocycle action (σ^{p}, v^{p}) can be perturbed to an actual action. Indeed, because we can take w_{g} = p, ∀g. Thus, if N^{σ} contains projections of any trace, then all 2-cocycle as in 3.1, obtained by reducing σ by projections, vanish.

However, we will see in the next section that for certain ergodic actions σ of property T groups, these cocycle actions cannot be perturbed to actions.

We now consider actions σ of groups G on type II_{1} factors N with the property that N is the core of a discrete decomposition of some type III factor (N , φ), in a way that the action σ itself be the restriction to N of an action on (N , φ). An example of actions satisfying this property is provided by the Connes-Stormer Bernoulli shifts defined in 2.2.3.

The next result provides a class of examples of non-trivial 1-cocycles for such actions. We’ll show in Section 4 that if G is infinite and has the property T, with σ being an action of G as in 2.2.3, then in fact these cocycles give the complete list of 1- cocycles for σ. But the construction may have an independent interest for general groups G as well.

To simplify notations, for each scalar 2-cocycle μ for G (see 1.2) we denote by F_{μ} the set of finite dimensional, unitary, projective representations of G with scalar 2-cocycle μ, i.e., π ∈ F_{μ} if π : G → U(n) for some n ≥ 0 and it satisfies π(g)π(h) = μ_{g,h}π(gh), ∀g, h ∈ G. The 0 representation is contained in F_{μ}, ∀μ. Note that if μ = 1 then F_{1} is just the set of finite dimensional representations F of G. This set contains the trivial representation of the group G and in fact any finite multiple of this representation.

Let (N , φ) be a von Neumann factor with a discrete decomposition, as in 2.2.0.

Let N = N_{φ} and H be the spectrum of this discrete decomposition, i.e., H is the multiplicative subgroup of ℝ_{+}^{∗} such that (N ⊂ N)\overline{⊗}B(ℓ^{2}(ℕ)) = (N⊗B(ℓ^{2}(ℕ)) ⊂ N⊗B(ℓ^{2}(ℕ)) ⋊ H). Let H_{1} = {β ∈ H | β ≤ 1}. We denote by F_{μ}^{H_{1}} the set of families of elements π_{β} in F_{μ} indexed by β ∈ H_{1} such that Σ_{β}n_{β}β ≤ 1, where n_{β} = dimπ_{β}. Note that some of the representations π_{β} may be equal to 0. In fact, if π_{1} ≠ 0 then this forces π_{1} to be a one dimensional representation, thus a character of G, and all other π_{β}, β < 1, must be zero.

Let G be an infinite group and σ an action of G on (N , φ), with its restriction to N = N_{φ} still denoted by σ.

3.2. Theorem. Let {π_{β}}_{β} ∈ F_{μ}^{H_{1}}. For each β let v_{i}^{β}, 1 ≤ i ≤ n_{β}, be isometries in V_{β} such that {v_{i}^{β}}_{β,i} have mutually orthogonal left supports. Let (π_{β}(g))_{ij} denote the coefficients of the representation π_{β} with respect to some orthonormal basis. Let w : G → N be defined by:

(3.2.1)  w_{g} = Σ_{β}Σ_{i,j}(π_{β}(g))_{ij}v_{i}^{β}σ_{g}(v_{j}^{β}^{∗}), g ∈ G

1^{∘}. w_{g} are partial isometries in N = N_{φ} and in fact w is a generalized weak 1-cocycle for σ with support p = Σ_{β,i}v_{i}^{β}v_{i}^{β}^{∗} and scalar 2-cocycle μ.

2^{∘}. Let σ′_{g} = Ad(w_{g}) ∘ σ_{g} and denote by B the von Neumann subalgebra of pN p generated by the matrix unit

{v_{i}^{β}v_{i′}^{β′}^{∗} | 1 ≤ i ≤ n_{β}, 1 ≤ i′ ≤ n_{β′}, β, β′ ∈ H_{1}}.

Let also B be the von Neumann subalgebra of B generated by

{v_{i}^{β}v_{i′}^{β}^{∗} | 1 ≤ i, i′ ≤ n_{β}, β ∈ H_{1}}.

Then B = B ∩ N = (B)_{φ} = E_{N}^{φ}(B) and σ′_{g}(B) = B, σ′_{g}(B) = B, with σ′_{g|B} ≃ Ad(π(g)), ∀g ∈ G, where π = ⊕_{β}π_{β} is viewed as a representation on the Hilbert space ⊕_{β}ℓ^{2}(n_{β}), with B being identified with the algebra B(⊕_{β}π_{β}) of all bounded operators on this Hilbert space.

3^{∘}. Assume in addition that σ is strongly mixing on N. Then (pN p)^{σ′} = B^{σ′} = π(G)′ ∩ B, (pNp)^{σ′} = B^{σ′} = π(G)′ ∩ B. Moreover, any finite dimensional vector subspace of pN p (resp. pNp) which is invariant to σ′ is contained in B (resp. B) and in fact B (resp. B) is the closure of the span of finite dimensional invariant subspaces of pN p (resp. pNp).

4^{∘}. Let w respectively w′ be cocycles constructed out of {π_{β}}_{β} ∈ F_{μ}^{H_{1}} and {v_{i}^{β}}_{β,i}, respectively {π′_{β}}_{β} ∈ F_{μ′}^{H_{1}} and {v′_{i}^{β}}_{β,i}, as in (3.2.1). If μ = μ′ and π_{β} ∼ π′_{β}, ∀β ∈ H_{1} then w and w′ are equivalent. If in addition we assume σ is strongly ergodic then, conversely, w ∼ w′ implies μ = μ′ and π_{β} ∼ π′_{β}, ∀β.

Proof. 1^{∘}. An easy calculation shows that if we denote p_{β} = Σ_{i}v_{i}^{β}^{∗}v_{i}^{β} then p_{β} is fixed by σ′_{g}, ∀g. Note also that p_{β} this way defined is in the center of B_{0}. Thus, by replacing w_{g} by p_{β}w_{g}, we see that in order to prove 1^{∘} in its full generality it is sufficient to prove it in the case all but one of the representations π_{β} are equal to zero, i.e., for π = π_{β}, for some β ∈ H_{1}.

Thus, we may assume w_{g} is of the form Σ_{i,j}(π(g))_{ij}v_{i}^{β}σ_{g}(v_{j}^{β}^{∗}). For simplicity, denote v_{i} = v_{i}^{β} and c_{ij}^{g} = (π(g))_{ij}. Taking into account that

Σ_{j}c_{ij}^{g}c_{jl}^{h} = μ_{g,h}c_{il}^{gh}

and that v_{i} have mutually orthogonal left supports, we get:

w_{g}σ_{g}(w_{h}) = (Σ_{i,j}c_{ij}^{g}v_{i}σ_{g}(v_{j}^{∗})σ_{g}(Σ_{k,l}c_{kl}^{h}v_{k}σ_{h}(v_{l}^{∗}))
= Σ_{i,l}(Σ_{j}c_{ij}^{g}c_{jl}^{h})v_{i}σ_{gh}(v_{l}^{∗})
= μ_{g,h}Σ_{i,l}c_{il}^{gh}v_{i}σ_{gh}(v_{l}^{∗}) = μ_{g,h}w_{gh}.

2^{∘}. Since E_{N}^{φ}(v′v^{∗}) = 0 for v ∈ V_{β} and v′ ∈ V_{β′} with β ≠ β′, we have E_{N}^{φ}(B) = B. To check that σ′_{g}(B) = B, we need to show that σ′_{g}(v_{i}^{β}v_{i′}^{β′}^{∗}) ∈ B, ∀i, i′, β, β′. We

have:

σ′_{g}(v_{i}^{β}v_{i′}^{β′}^{∗}) = w_{g}σ_{g}(v_{i}^{β}v_{i′}^{β′}^{∗})w_{g}^{∗}
= (Σ_{α,k,l}(π_{α}(g))_{kl}v_{k}^{α}σ(v_{l}^{α}^{∗}))σ_{g}(v_{i}v_{i′}^{β′}^{∗})(Σ_{α′,k′,l′}(π_{α′}(g))_{k′l′}v_{k′}^{α′}σ_{g}(v_{l′}^{α′}^{∗}))^{∗}
= Σ_{k,k′}(π_{β}(g))_{ki}\overline{π_{β′}(g)_{k′i′}}v_{k}^{β}σ_{g}(v_{i}^{β}^{∗}))σ_{g}(v_{i}^{β}v_{i′}^{β′}^{∗})σ_{g}(v_{i′}^{β′})v_{k′}^{β′}^{∗}
= Σ_{k,k′}(π_{β}(g))_{ki}\overline{(π_{β′}(g))_{k′i′}}v_{i′}v_{k′}^{β′}^{∗}.

Since this latter element lies in B, we are done. The above computation also shows that if we take β = β′ then σ′(v_{i}^{β}v_{i′}^{β}^{∗}) lies in sp{v_{k}^{β}v_{k′}^{β}^{∗}}_{k,k′}, thus in B.

Moreover, it shows that if we identify the von Neumann algebra B generated by the matrix unit {v_{i}^{β}v_{i′}^{β′}^{∗} | i, i′, β, β′} with B(⊕_{β}ℓ^{2}(n_{β})), then for each x = v_{i}^{β}v_{i′}^{β′}^{∗} ∈ B we have σ′_{g}(x) = π(g)xπ(g)^{∗}.

3^{∘}. Recall that

w_{g} = Σ_{α}Σ_{k,l}(π_{α}(g))_{k,l}v_{k}^{α}σ_{g}(v_{l}^{α}^{∗}).

Hence

w_{g}^{∗} = Σ_{α′}Σ_{k′,l′} \overline{(π_{α′}(g))_{k′,l′}}σ_{g}(v_{l′}^{α′})v_{k′}^{α′}^{∗}

Any element x ∈ B can be expressed in the form

x = Σ_{β,β′}Σ_{i,i′}v_{i}^{β}x_{ii′}^{ββ′}v_{i′}^{β′}^{∗},

for some x_{ii′}^{ββ′} ∈ B. For such x we have:

σ_{g}(x) = Σ_{β,β′}Σ_{i,i′}σ_{g}(v_{i}^{β})σ_{g}(x_{ii′}^{ββ′})σ_{g}(v_{i′}^{β′}^{∗})

By taking into account that v_{l}^{α}^{∗}v_{i}^{β} = δ_{αβ}δ_{li} and v_{i′}^{β′}^{∗}v_{l′}^{α′} = δ_{β′α′}δ_{i′l′} it follows that σ_{g}(v_{l}^{α}^{∗})σ_{g}(v_{i}^{β}) = δ_{αβ}δ_{li} and σ_{g}(v_{i′}^{β′}^{∗})σ_{g}(v_{l′}^{α′}) = δ_{β′α′}δ_{i′l′} as well. Thus we have:

(3.2.2)  w_{g}σ_{g}(x)w_{g}^{∗} = Σ_{β,β′}Σ_{k,k′}Σ_{i,i′}(π_{α}(g))_{ki}\overline{(π_{β′}(g)_{k′i′}}v_{k}^{β}σ_{g}(x_{ii′}^{ββ′})v_{k′}^{β′}^{∗}

The equation w_{g}σ_{g}(x)w_{g}^{∗} = x is then equivalent to the set of equations

(3.2.3)  Σ_{i,i′}(π_{α}(g))_{ki}\overline{(π_{β′}(g)_{k′i′}}v_{k}^{β}σ_{g}(x_{ii′}^{ββ′})v_{k′}^{β′}^{∗} = x_{kk′}^{ββ′}, ∀i, i′, β, β′

Letting y_{ii′}^{ββ′} = σ_{g}(x_{ii′}^{ββ′}) and x_{ii′}^{ββ′} = σ_{g^{−1}}(y_{ii′}^{ββ′}), this shows in particular that the finite dimensional space sp{y_{ii′}^{ββ′} | 1 ≤ i ≤ n_{β}, 1 ≤ i′ ≤ n_{β′}} is invariant to σ_{g^{−1}}, ∀g ∈ G, thus to σ_{h}, ∀h ∈ G. Since σ is strongly mixing on N, by 2.4.3 this implies y_{ii′}^{ββ′} are all scalars. Thus, x_{ii′}^{ββ′} are scalars as well implying that x lies in B. By part 2^{∘} it follows that x ∈ B^{σ′} = π(G)′ ∩ B. Restricting to elements x ∈ pNp, we also get (pNp)^{σ′} = B^{σ′} = π(G)′ ∩ B.

Let now H_{0} ⊂ pN p be a finite dimensional vector subspace invariant to σ′. Since the projection p_{β}Σ_{i}v_{i}^{β}v_{i}^{β}^{∗} are fixed by σ′, ∀β, it follows that

(3.2.4)  X = p_{β}H_{0}p_{β′} = Σ_{ii′}v_{i}^{β}X_{ii′}^{ββ′}v_{i′}^{β′}^{∗}

is invariant to σ′ as well, where X_{ii′}^{ββ′} = v_{i}^{β}^{∗}H_{0}v_{i′}^{β′}.

But the calculation (3.2.2) above shows that

(3.2.5)  w_{g}σ_{g}(X)w_{g}^{∗} = Σ_{k,k′}Σ_{i,i′}(π_{α}(g))_{ki}\overline{(π_{β′}(g)_{k′i′}}v_{k}^{β}σ_{g}(X_{ii′}^{ββ′})v_{k′}^{β′}^{∗}

Equating (3.2.4) with (3.2.5) and letting Y_{ii′}^{ββ′} = σ_{g}(X_{ii′}^{ββ′}), X_{ii′}^{ββ′} = σ_{g^{−1}}(Y_{ii′}^{ββ′}), it follows that the finite dimensional vector space sp{Y_{ii′}^{ββ′} | 1 ≤ i ≤ n_{β}, 1 ≤ i′ ≤ n_{β′}} is invariant to σ_{g^{−1}}, ∀g ∈ G, thus to σ_{h}, ∀h ∈ G.

By 2.4.3 it follows again that all the spaces Y_{ii′}^{ββ′} are equal to ℂ. Thus X_{ii′}^{ββ′} = ℂ as well, implying that X = Σ_{i,i′}ℂv_{i}^{β}v_{i′}^{β′}^{∗}. This proves that H_{0} ⊂ B.

Since p_{β}Bp_{β′} are all invariant subspaces and they span B, the last part of the statement follows as well.

4^{∘}. Note first that the equivalence class of the cocycle w does not depend on the choice of the isometries v_{i}^{β}, once the family of representations π_{β} is fixed.

If the two given families of representations satisfy π_{β} ∼ π′_{β}, ∀β ∈ H_{1}, then in particular dim(π_{β}) = dim(π′_{β}), ∀β. It follows that in the contruction of w and w′ we can take the same set of isometries {v_{i}^{β}}_{β,i}. Thus, both π_{β}, π′_{β} “live” in p_{β}Bp_{β},

where p_{β} = Σ_{i}v_{i}^{β}v_{i}^{β}^{∗} as usual.

If for each β ∈ H_{1} with n_{β} ≠ 0 we take u_{β} ∈ p_{β}Bp_{β} to be the unitary element that implements the equivalence of π_{β}, π′_{β} then an immediate calculation shows that u = ⊕_{β}u_{β} also implements the equivalence of w with w′.

Note that this same computation shows that the converse also holds true, provided dim(π_{β}) = dim(π′_{β}). But this equality does hold true if we assume σ is strongly ergodic, by the last part of 3^{∘}. Indeed, because by that part we have that the fixed point algebras B, respectively B′, of the actions Ad(w)∘σ and respectively Ad(w′) ∘ σ, are conjugate in pNp. Since the traces of the minimal projections in p_{β}Bp_{β} and p′_{β}B′p′_{β} must both be equal to β, thus being different for distinct β, it follows that the the unitary element u ∈ pNp that satisfies uBu^{∗} = B′ must carry projections of trace β onto projections of trace β, implying that up_{β}Bp_{β}u^{∗} = p′_{β}B′p′_{β} and thus n_{β} = n′_{β}, ∀β. Q.E.D.

An immediate consequence of Theorem 3.2 is that any cocycle action constructed by reduction-amplification of an action σ as in 3.2, but with the factor N being of type II, can in fact be perturbed to an action. We will later see that in case N is of type II_{1}, then this may not be the case.

3.3 Corollary. With σ as in 3.2 above, assume in addition that H_{1} ≠ {1} (equivalently, the factor (N , φ) is of type III). For each t > 0, let σ^{t} be a choice of a cocycle action on N^{t} obtained by reducing, or amplifying the action σ on N = N_{φ}, as in 3.1.. Then σ^{t} can be perturbed to an action.

Proof. Let m be an integer such that m ≥ t. Since H_{1} ≠ {1}, it contains some β < 1. Thus, β^{n} ∈ H_{1}, ∀n ≥ 1 as well. But then there exist some non-negative integers k_{n} such that Σ_{n}k_{n}β^{n} = t/m and we can apply Theorem 3.2 to the action σ ⊗ id_{m} of G on N ⊗ M_{m×m}(ℂ) (which is the core of the discrete decomposition of (N ⊗ M_{m×m}(ℂ), φ ⊗ tr)), to provide a generalized 1-cocycle w for σ with support p, where p ∈ N ⊗ M_{m×m}(ℂ) is a projection of trace t/m. Q.E.D.

Our next result gives an abstract characterization of the cocycles w constructed in the previous Theorem. Namely, we show that w is locally of the form (3.2.1) iff the projective representation ξ ↦ σ_{g}(ξ)w_{g}^{∗} has finite dimensional invariant subspaces. This observation will be needed in the next Section.

3.4. Proposition. With the same notations and hypothesis as in 3.2, assume the action σ : G → Aut(N , φ) is strongly mixing. Let w be a generalized weak 1-cocycle for σ with support p ∈ P(N) and scalar 2-cocycle μ.

1^{∘}. For g ∈ G and ξ ∈ L^{2}(N p, φ) denote σ_{g}^{w}(ξ) = σ_{g}(ξ)w_{g}^{∗}. Then σ^{w} is a projective unitary representation of G on L^{2}(N p, φ) with scalar 2-cocycle μ.

2^{∘}. Let H be the Hilbert space of Hilbert-Schmidt operators on L^{2}(N p, φ). For each X ∈ H let σ̃_{g}^{w}(X) = σ_{g}^{w}Xσ_{g}^{w}^{∗}. Then σ̃^{w} is a unitary representation of G on H.

3^{∘}. The following conditions are equivalent:

(i). σ̃^{w} contains a copy of the trivial representation.

(ii). σ^{w} has a non-trivial, invariant, finite dimensional subspace H_{0} ⊂ L^{2}(N p, φ). (iii). There exist some β ∈ H_{1} and a finite set of isometries v_{1}, v_{2}, ..., v_{n} ∈ V_{β}

with mutually orthogonal left supports such that H_{0} = Σ_{i}ℂv_{i}^{∗} is invariant to σ^{w} and q = Σ_{i}v_{i}v_{i}^{∗} ≤ p is fixed by the action σ′ = Adw ∘ σ.

(iv). There exists a non-zero projection q ∈ pNp fixed by σ′ such that qw is of the form qw_{g} = Σ_{i,j}(π_{0}(g))_{ij}v_{i}σ_{g}(v_{j}^{∗})), g ∈ G, for some π_{0} ∈ F_{μ} and some isometries v_{i}, 1 ≤ i ≤ dim(π_{β}), lying all in some V_{β} and having mutually orthogonal left supports.

(v). There exists a non-zero projection q_{0} ∈ pNp fixed by σ′ such that q_{0}w is of the form (3.2.1), for some family {π_{β}}_{β} ∈ F_{μ}^{H_{1}}, and such that there are no non-zero projections q ≤ (p − q_{0}), q ∈ B_{0} with the property that qw is of the form (3.2.1).

Proof. 1^{∘}. By the definitions, σ_{g}^{w} are clearly unitary operators acting on L^{2}(N p, φ). Also, we have

σ_{g}^{w}(σ_{h}^{w}(ξ)) = σ_{g}^{w}(σ_{h}(ξ)w_{h}^{∗})
= σ_{g}(σ_{h}(ξ)w_{h}^{∗})w_{g}^{∗} = σ_{gh}(ξ)σ_{g}(w_{h}^{∗})w_{g}^{∗}
= μ_{g,h}σ_{gh}(ξ)w_{gh}^{∗}.

proving 1^{∘}.

2^{∘}. By 1^{∘} it follows that

σ̃_{g}^{w}(σ̃_{h}^{w}(X)) = σ_{g}^{w}σ_{h}^{w}Xσ_{h}^{w}^{∗}σ_{g}^{w}^{∗}

= μ_{g,h}\overline{μ_{g,h}}σ_{gh}^{w}Xσ_{gh}^{w}^{∗} = σ̃_{gh}^{w}(X).

3^{∘}. Noticing that σ̃^{w} extends from the ideal of Hilbert-Schmidt operators to all B(L^{2}(N p, φ)), it follows that if X ∈ H is a fixed point for σ̃^{w} then the trace class operator X^{∗}X is also a fixed point for σ̃^{w}. Thus all spectral projections P of X^{∗}X, are fixed points for σ̃^{w} as well. Since P have finite trace, they are finite dimensional. Let H_{0} ⊂ L^{2}(N p, φ) be a non-zero finite dimensional space corresponding to some spectral projection P ≠ 0. By the definitions, σ̃_{g}^{w}(P) is the orthogonal projection onto σ_{g}^{w}(H_{0}). Since σ̃_{g}^{w}(P) = P, it follows that σ_{g}^{w}(H_{0}) = H_{0}. This proves (i) ⟹ (ii).

To prove (ii) ⟹ (iii), note first that if v ∈ V then each of the subspaces L^{2}(v^{∗}Np, φ) and L^{2}(Nvp, φ) of L^{2}(N , φ) are invariant to σ^{w}. Thus, if σ^{w} has a non-trivial finite dimensional invariant subspace H_{0} then by compressing it to one of these spaces and taking into account that they span all L^{2}(N p, φ) it follows that we may assume H_{0} is contained either in some L^{2}(v^{∗}Np, φ) or in some L^{2}(Nvp, φ).

In either case, denote by {ξ_{i}}_{i} an orthonormal basis of H_{0}. By the definition of σ^{w} it follows that Σ_{i}ξ_{i}ξ_{i}^{∗} ∈ L^{1}(N , φ) is fixed by σ. Since σ is strongly mixing, it is ergodic. Thus Σ_{i}ξ_{i}ξ_{i}^{∗} ∈ ℂ1 implying that ξ_{i} are actually in N p.

Similarily, the finite dimensional vector space H′_{0} =spH_{0}H_{0}^{∗} ⊂ N is invariant to σ. It follows that H′_{0} = ℂ implying that ξ_{i}ξ_{j}^{∗} ∈ ℂ1, ∀i, j. This in turn implies that each ξ_{i}^{∗} is a scalar multiple of an isometry v_{i} and that A = sp{v_{i}v_{j}^{∗} | 1 ≤ i, j ≤ n} is a finite dimensional factor. Thus, by replacing if necessary the elements {v_{i}}_{i} by some elements {a_{j}v_{j}}_{j} for some appropriate partial isometries a_{j} ∈ A, we may assume the isometries v_{i} have mutually orthogonal left supports, yet still generate H_{0}. Since all elements in L^{2}(Nvp, φ) have left supports ≠ 1 while the left support of ξ_{i} = v_{i}^{∗} is 1, it follows that H_{0} cannot be a subspace of some L^{2}(Nvp, φ), forcing it to be a susbspace of some L^{2}(v^{∗}Np, φ), v ∈ V. This implies v_{i} ∈ V_{β}, ∀i, where β = φ(vv^{∗}) ∈ H_{1}. Moreover, since spH_{0}^{∗}H_{0} is invariant to

pN p ∋ x ↦ w_{g}σ_{g}(x)w_{g}^{∗} = σ′_{g}(x)

and the support projection q of the elements in this vector space is Σ_{i}v_{i}v_{i}^{∗}, it follows that σ′_{g}(q) = q, ∀g ∈ G.

Assuming (iii) holds true, let β ∈ H_{1} and v_{1}, v_{2}, ..., v_{n} ∈ V_{β} be such that H_{0} = Σ_{i}ℂv_{i}^{∗} is invariant to σ^{w}. It follows that for each 1 ≤ i ≤ n and g ∈ G there exist some scalars {b_{ij}^{g}}_{i,j} such that σ_{g}(v_{i}^{∗})w_{g}^{∗} = Σ_{j}b_{ij}^{g}v_{j}^{∗}, ∀i.

Denote by c_{ij}^{g} = \overline{b_{ji}^{g}}. We have to prove that c_{ij}^{g} are the coefficients of a projective unitary representation of G with the same scalar 2-cocycle μ as w.

Since σ_{g}(v_{i}^{∗})w_{g}^{∗} = Σ_{j}b_{ij}^{g}v_{j}^{∗}, ∀i, we have qw_{g} = Σ_{i,j}c_{ji}^{g}v_{j}σ_{g}(v_{i}^{∗}). Similarily qw_{h} = Σ_{i,j}c_{ji}^{h}v_{j}σ_{h}(v_{i}^{∗}). It follows that

w_{g}σ_{g}(w_{h}) = (Σ_{i,j}c_{ji}^{g}v_{j}σ_{g}(v_{i}^{∗}))(Σ_{k,l}c_{lk}^{h}σ_{g}(v_{l})σ_{gh}(v_{k}^{∗}))
= Σ_{j,k}(Σ_{i}c_{ji}^{g}c_{ik}^{h})v_{j}σ_{gh}(v_{k}^{∗}).

Smilarily,we have

w_{gh} = Σ_{j,k}c_{jk}^{gh}v_{j}σ_{gh}(v_{k}^{∗}).

Replacing the above in the equation w_{g}σ_{h}(w_{g}) = μ_{g,h}w_{gh} and multiplying on the left by v_{j}v_{j}^{∗} and to the right by v_{k}v_{k}^{∗}, we get

Σ_{i}c_{ji}^{g}c_{ik}^{h} = μ_{g,h}c_{j,k}^{gh}, ∀j, k, g.

But this means c_{ij}^{g} are the coefficients of a projective representation π_{0} of G on ℓ^{2}(n) with 2-cocycle μ.

To prove (iv) ⟹ (v) we use a maximality argument. Thus, we let S denote the set of families of mutually orthogonal projections (q^{i})_{i} ⊂ pNp such that q^{i} ∈ (pNp)^{σ′} and q^{i}w is of the form (3.2.1). We endow S with the obvious order ≤ given by inclusion. (S, ≤) this way defined is clearly inductively ordered. Let (q_{0}^{i})_{i} be a maximal element, which by (iv) we may suppose non-zero. Let q_{0} = Σ_{i}q_{0}^{i} ∈ (pNp)^{σ′}. Then q_{0} clearly satisfies the required conditions, or else the maximality of (q_{i}^{0})_{i} would be contradicted.

The implication (v) ⟹ (iv) is trivial. Finally, (iv) ⟹ (i) follows by taking P ∈ H to be the orthogonal projection of L^{2}(N p, φ) onto H_{0} = Σ_{i}ℂv_{i}^{∗}. Indeed, because by hypothesis we have

σ_{g}(v_{i}^{∗})w_{g}^{∗} = Σ_{j}\overline{(π_{0}(g))_{ij}}v_{j}^{∗}

showing that σ_{g}^{w}(H_{0}) = H_{0}, equivalently σ̃_{g}^{w}(P) = P, ∀g ∈ G. Q.E.D.

4. Cohomology of Connes-Størmer Bernoulli shifts.

Let N_{0} = M_{k×k}(ℂ) and φ_{0} be a faithful state on N_{0}. Let G be an infinite property T group. Let (N , φ) = \overline{⊗_{g∈G}}(N_{g}, φ_{g}), where (N_{g}, φ_{g}) = (N_{0}, φ_{0}), ∀g ∈ G.

As explained in 2.2.0, the ITPF1 factor (N , φ) has a discrete decomposition over N = N_{φ}, with the type II_{1} factor N being isomorphic to the hyperfinite factor.

We denote by σ both the (N_{0}, φ_{0})-Bernoulli shift of G on (N , φ) and the Connes-Størmer Bernoulli shift obtained by restricting this action to N. In Theorem 3.2 of the previous section we constructed a family of generalized weak 1-cocycles for such Connes-Størmer Bernoulli shifts σ. In this section we prove that the cocycles in 3.2 give a complete list of cocycles for σ. The notations V, V_{β}, H_{1}, F_{μ}, F_{μ}^{H_{1}} are the ones used in 2.2.0 and 3.2.

4.1. Theorem. Let w be a generalized weak 1-cocycles of the Connes-Størmer Bernoulli shift σ, with support p and 2-cocycle μ. Let B_{0} be the fixed point algebra of the action σ′_{g} = Ad(w_{g}) ∘ σ_{g}, g ∈ G, on pNp. Then we have:

1^{∘}. If z ∈ B_{0} is the maximal central projection of B_{0} such that B_{0}z is atomic, then there exist a family of finite dimensional, projective, unitary representations {π_{β} | β ∈ H_{1}} ⊂ F_{μ}^{H_{1}} of the property T group G and some isometries {v_{i}^{β} | 1 ≤ i ≤ dimπ_{β}} ⊂ V_{β}, β ∈ H_{1}, such that Σ_{β,i}v_{i}^{β}v_{i}^{β}^{∗} = z and such that the generalized weak 1-cocycle zw is given by the formula:

(4.1.1)  zw_{g} = Σ_{β}(Σ_{i,j}(π_{β}(g))_{ij}v_{i}^{β}σ_{g}(v_{j}^{β}^{∗})), g ∈ G

2^{∘}. Assume in addition that (N_{0}, φ_{0}) = ⊗_{i=1}^{k}(N_{0,i}, φ_{0,i}), where N_{0,i} ≃ M_{2×2}(ℂ)

and φ_{0,i} are faithful states on N_{0,i}, 1 ≤ i ≤ k. Then B_{0} is atomic, i.e., z = 1_{B_{o}} = p, so the above formula (4.1.1) holds true for the given cocycle w(= pw) itself.

To prove the theorem we first need some lemmas. To state them, recall some notations from 2.2.1-2.2.2. Thus, we consider the generalized (N_{0} ⊗ N_{0}, φ_{0} ⊗ φ_{0}) -Bernoulli shift of G on N \overline{⊗}N which we identify in the obvious way with σ ⊗σ. We denote by θ the associated Connes-Størmer Bernoulli shift of G on the centralizer Ñ of φ̃ = φ ⊗ φ on N \overline{⊗}N. Thus, θ_{g} = (σ_{g} ⊗ σ_{g})_{|Ñ}, g ∈ G.

We denote by α : ℝ → AutÑ the product type action considered in 1∘-3∘ of 2.5.1. Note that α commutes with the Bernoulli shift action θ defined above. Thus, θ and α implement an action θ ⊕ α of the group G ⊕ ℝ on Ñ.

For the next two lemmas, the discrete group G can be arbitrary. Other than that, we are under the general assumptions of 4.1.

4.2. Lemma. Let M denote the type II_{1} factor obtained by taking the cross product of Ñ by the action θ ⊕ α of the group G = G ⊕ ℝ on it, in which ℝ is regarded as a discrete group. Let (U_{h})_{h∈G} ⊂ M be the canonical unitaries implementing this action. Let also U′_{g} = w_{g}U_{g}, g ∈ G, and P = {U′_{g}}′′_{g∈G} ⊂ pMp be the factor with support 1_{P} = p they generate in pMp. Fix x ∈ N = N ⊗ ℂ. For each t ∈ ℝ put

x_{t} =^{def} E_{P′∩pMp}(pxU_{t}p)U_{t}^{∗}. Then we have:

1^{∘}. x_{t} belongs to Ñ being in fact the unique element of minimal norm-2 in the weakly closed convex subset of Ñ

K′_{t}(x) = \overline{co}^{w}\{U′_{g}xU_{t}U′_{g}^{∗}U_{t}^{∗}\}_{g∈G} = \overline{co}^{w}\{w_{g}σ_{g}(x)α_{t}(w_{g}^{∗})\}_{g∈G}.

2^{∘}. x_{t} satisfies the equivalent conditions

(a)  U′_{g}x_{t} = x_{t}AdU_{t}(U′_{g}), ∀g ∈ G

(b)  w_{g}θ_{g}(x_{t}) = x_{t}α_{t}(w_{g}), ∀g ∈ G

Also, x_{t} is the unique element in K′_{t}(x) that satisfies these equivalent conditions. 3^{∘}. x_{t}x_{t}^{∗} ∈ B_{0} ⊗ ℂ and x_{t}^{∗}x_{t} ∈ α_{t}(B_{0} ⊗ ℂ). 4^{∘}. p_{t^{∗}} = α_{−t}(p_{t})^{∗} and (bx)_{t} = b(x_{t}), ∀b ∈ B_{0}.

Proof. Since the set {U′_{g}}_{g∈G} is total in P, E_{P′∩pMp}(xU_{t}) is the unique element of minimal norm-2 in K_{t}(x) = \overline{co}^{w}\{U′_{g}xU_{t}U′_{g}^{∗}\}_{g∈G} (see e.g., [Po4]). But U′_{g}xU_{t}U′_{g}^{∗} = w_{g}σ_{g}(x)U_{t}w_{g}^{∗} = w_{g}σ_{g}(x)α_{t}(w_{g}^{∗})U_{t}, implying that K′_{t}(x) = K_{t}(x)U_{t}^{∗} and that x_{t} = E_{P′∩pMp}(xU_{t})U_{t}^{∗} is the unique element of minimal norm-2 in the set K′_{t}(x). This proves 1^{∘}.

The commutation relation U′_{g}(x_{t}U_{t}) = x_{t}U_{t}U′_{g}, which holds true for all g ∈ G, is equivalent to the condition U′_{g}x_{t} = x_{t}AdU_{t}(U′_{g}) (by multiplying the former to the right by U_{t}^{∗}). This shows that (a) in 2^{∘} holds true. But condition (a) amounts to w_{g}U_{g}x_{t} = x_{t}U_{t}w_{g}U_{g}U_{t}^{∗}, which multiplied from the right by U_{g}^{∗} gives condition (b) (after apropriate simplifications).

Moreover, since

uE_{P′∩pMp}(y)u^{∗} = E_{P′∩pMp}(uyu^{∗}), ∀u ∈ P, y ∈ pMp,

it follows that E_{P′∩pMp}(y) = E_{P′∩pMp}(pxU_{t}p), ∀y ∈ K_{t}(x), so that x_{t}U_{t} is the unique element in K_{t}(x) = K′_{t}(x)U_{t} that commutes with all U′_{g}, g ∈ G. By the equivalence between the commutation relation and the conditions (a) and (b), this proves the uniqueness in 2^{∘}.

Since by the way it is defined the element x_{t}U_{t} commutes with the ^{∗}-algebra P, 2.4.2 implies that

x_{t}x_{t}^{∗} = (x_{t}U_{t})(x_{t}U_{t})^{∗} ∈ P′ ∩ pÑp
= (pÑp)^{θ′} = (pNp)^{σ′} ⊗ ℂ = B_{0} ⊗ ℂ,

where θ′ is the action of G on pÑp given by θ′_{g} = σ′_{g} ⊗ σ_{g}, g ∈ G.

Similarly, we get

α_{−t}(x_{t}^{∗}x_{t}) = (x_{t}U_{t})^{∗}(x_{t}U_{t}) ∈ P′ ∩ pÑp = B_{0} ⊗ ℂ,

implying that x_{t}^{∗}x_{t} ∈ α_{t}(B_{0} ⊗ ℂ). This proves 3^{∘}.

Taking −t for t, by the definitions we get p_{−t}U_{−t} = E_{P′∩pMp}(pU_{−t}p). Thus

U_{t}p_{−t}^{∗}U_{t}^{∗} = E_{P′∩pMp}(pU_{t}p)U_{−t} = p_{t},

showing that α_{t}(p_{−t})^{∗} = p_{t}, which by taking adjoints and applying (α_{t})^{−1} = α_{−t} gives the first part of 4^{∘}. The second part is trivial by the definitions. Q.E.D.

4.3. Lemma. Let u(t) = (p_{t}p_{t}^{∗})^{−1/2}p_{t} ∈ Ñ denote the partial isometry in the polar decomposition of p_{t} and denote by l(t) = u(t)u(t)^{∗}, r(t) = u(t)^{∗}u(t) its left support and right support. Then we have:

1^{∘}. U′_{g}u(t) = u(t)U_{t}U′_{g}U_{t}^{∗}, ∀g ∈ G.

2^{∘}. u(−t) = α_{−t}(u(t))^{∗}.

3^{∘}. l(t) ∈ B_{0}, r(t) ∈ α_{t}(B_{0}) and u(t)^{∗}B_{0}u(t) = r(t)α_{t}(B_{0})r(t).

Proof. By multiplying the relation 4.2.2^{∘} by (p_{t}p_{t}^{∗})^{−1/2} ∈ B_{0} from the left, we get 2^{∘}.

Since by 4.2.4^{∘} we have p_{−t} = α_{−t}(p_{t})^{∗}, by the definitions of u(t) and u(−t) we immediately get u(−t) = α_{−t}(u(t))^{∗}. This proves 2^{∘}.

Taking (p_{t}p_{t}^{∗})^{−1/2} to be the element b in 4.2.4^{∘}, by 4.2.3∘ and the definitions we get l(t) = u(t)u(t)^{∗} = b_{t}b_{t}^{∗} ∈ B_{0} and r(t) = u(t)^{∗}u(t) = b_{t}^{∗}b_{t} ∈ α_{t}(B_{0}).

Similarily, if we let q be an arbitrary projection in l(t)B_{0}l(t) and take this time q(p_{t}p_{t}^{∗})^{−1/2} to be the element b in 4.2.4∘, then we get u(t)^{∗}qu(t) = b_{t}^{∗}b_{t} ∈ α_{t}(B_{0}). Since any element in B can be approximated in uniform norm by a linear combination of projections, it follows that u(t)^{∗}B_{0}u(t) ⊂ r(t)α_{t}(B_{0})r(t), proving one inclusion in 1^{∘}.

Taking −t for t in this inclusion, we also have u(−t)B_{0}u(−t)^{∗} ⊂ r(−t)α_{−t}(B_{0})r(−t). Applying α(t) on both sides we get:

α_{t}(u(−t))α_{t}(B_{0})α_{t}(u(−t)^{∗}) ⊂ α_{t}(r(−t))B_{0}α_{t}(r(−t)).

But by part 2∘ we have α_{t}(u(−t)) = u(t)^{∗} and l(t) = l(α_{t}(u(−t))^{∗}) = α_{t}(r(−t)), so the above implies

u(t)^{∗}α_{t}(B_{0})u(t) ⊂ l(t)B_{0}l(t).

Equivalently,

α_{t}(B_{0}) ⊂ u(t)^{∗}B_{0}u(t).

This proves the opposite inclusion in 3^{∘}. Q.E.D.

For the next three lemmas we will assume that the group G has the property T, as in the hypothesis of 4.1. All other assumptions and notations are as in 4.1-4.3.

4.4. Lemma. For each b ∈ B_{0}, the function t ↦ b_{t} from ℝ into the unit ball of Ñ ⊂ M is continuous at t = 0, with respect to the norm-2 topology. Moreover,

lim_{t→0}∥u(t) − p∥_{2} = 0.

Proof. By 4.2.4^{∘}, to prove the continuity at t = 0 of the function t ↦ b_{t} it is sufficient to prove it for the function t ↦ p_{t}.

Since t ↦ α_{t} is pointwise continuous in the norm-2 topology, there exists ε_{1} > 0 such that if t ∈ ℝ satisfies |t| ≤ ε_{1} then

∥w_{g_{i}}α_{t}(w_{g_{i}})^{∗} − pα_{t}(p)∥_{2} ≤ ε_{0}∥p − pα_{t}(p)∥_{2}, ∀1 ≤ i ≤ n.

Consider the representation π of G on L^{2}(pMp) given by π(g)ξ = U′_{g}ξU′_{g}^{∗}. Then for t ∈ ℝ and ξ = pU_{t}p, we have

∥π(g)ξ − ξ∥ = ∥U′_{g}U_{t}U′_{g}^{∗} − pU_{t}p∥_{2}
= ∥w_{g}U_{v}w_{g}^{∗} − pU_{t}p∥_{2} = ∥w_{g}α_{t}(w_{g})^{∗} − pα_{t}(p)∥_{2}, ∀g ∈ G.

Applying this to g = g_{i}, i = 1, 2, ..., n, we get

∥π(g_{i})ξ − ξ∥ = ∥w_{g_{i}}α_{t}(w_{g_{i}})^{∗} − pα_{t}(p)∥_{2}.

By Lemma A.1 it thus follows that

∥w_{g}α_{t}(w_{g})^{∗} − pα_{t}(p)∥_{2} ≤ Kmax_{i}∥w_{g_{i}}α_{t}(w_{g_{i}})^{∗} − pα_{t}(p)∥_{2}, ∀g ∈ G.

By part 2^{∘} it follows that

∥p_{t} − pα_{t}(p)∥_{2} ≤ Kmax_{i}∥w_{g_{i}}α_{t}(w_{g_{i}})^{∗} − pα_{t}(p)∥_{2}

forall t ∈ ℝ with |t| ≤ ε_{1}. In particular, since p_{0} = p and ∥pα_{t}(p) − p∥_{2} → 0 as t → 0, this implies the continuity at t = 0 of the function p_{t}, finishing the proof of the statement.

The continuity of u(t) at t = 0 follows trivially from the first part and 4.3. Q.E.D.

4.5. Lemma. Let z_{0} ∈ B_{0} be a central projection of B_{0} such that B_{0}z_{0} is a finite dimensional algebra. Then there exists ε > 0 such that if |t| < ε then z_{0} ≤ u(t)u(t)^{∗} and u(t)^{∗}B_{0}z_{0}u(t) = α_{t}(B_{0}z_{0}).

Proof. By Lemma 4.4 we have lim_{t→0}u(t)u(t)^{∗} = 1_{B_{0}} = p. Since z_{0} is central in B_{0}

and B_{0}z_{0} is a finite von Neumann algebra, it follows that for |t| small enough we have z_{0} ≤ u(t)u(t)^{∗}.

Let c > 0 be the minimal trace of a non-zero projection in B_{0}z_{0}. Note that for |t| small enough, α_{t}(z_{0}) is close to z_{0}. By 4.4, u(t)^{∗}z_{0}u(t) is also close to z_{0}. Thus, by 4.3 it follows that α_{t}(z_{0}) and u(t)^{∗}z_{0}u(t) are projections of α_{t}(B_{0}) which are close one to another for |t| small. Moreover, α_{t}(z_{0}) is central in α_{t}(B_{0}).

Since the minimal trace of a non-zero projection in both

u(t)^{∗}(B_{0}z_{0})u(t) = u(t)^{∗}z_{0}u(t)α_{t}(B_{0})u(t)^{∗}z_{0}u(t)

and in

α_{t}(B_{0}z_{0}) = α_{t}(B_{0})α_{t}(z_{0})

is c, it follows that if ∥u(t)^{∗}z_{0}u(t) − α_{t}(z_{0})∥_{2}^{2} < c then u(t)^{∗}z_{0}u(t) = α_{t}(z_{0}). Thus, for |t| small enough we have u(t)^{∗}B_{0}z_{0}u(t) = α_{t}(B_{0}z_{0}). Q.E.D.

4.6. Lemma. Let z ∈ B_{0} be the maximal atomic projection of B_{0}, as in the statement of Theorem 4.1.. There exists a partial isometry u ∈ Ñ such that uu^{∗} = z ∈ B_{0}, u^{∗}u = α_{1}(z) ∈ α_{1}(B_{0}) and

(w_{g} ⊗ 1)U_{g}u = u(1 ⊗ w_{g})U_{g}, ∀g ∈ G.

Proof. Let {z_{i}}_{i} be a partition of z with central projections in B_{0} such that each B_{0}z_{i} is finite dimensional. It is suficient to prove that for each z_{i} there exists a partial isometry u_{i} ∈ Ñ such that z_{i} = u_{i}u_{i}^{∗} and (w_{g} ⊗1)U_{g}u_{i} = u_{i}(1⊗w_{g})U_{g}, ∀g ∈ G. Indeed, because then we can just define u = ⊕_{i}u_{i} and use the fact that for this u we have q_{i}(w_{g} ⊗ 1)U_{g}u = (w_{g} ⊗ 1)U_{g}u_{i}. Summing up this gives (w_{g} ⊗ 1)U_{g}u = u(1 ⊗ w_{g})U_{g}.

This shows that by replacing w by z_{i}w and B_{0} by B_{0}z_{i} we may suppose B_{0} is finite dimensional. Denote z_{0} = 1_{B_{0}}. By 4.5 it follows that there exists n ≥ 1 such that u(t_{n})^{∗}z_{0}u(t_{n}) = α_{t_{n}}(z_{0}), where t_{n} = 1/2^{n}.

By applying recursively (α_{t_{n}})^{k} to this equation, for k = 1, 2, ..., n − 1, it follows that the element

u = u(t_{n})(α_{t_{n}}(u_{t_{n}}))(α_{t_{n}}^{2}(u(t_{n})))...(α_{t_{n}}^{n−1}((u(t_{n})))

is a partial isometry in Ñ which satisfies u^{∗}z_{0}u = α_{t_{n}}^{n}(z_{0}) = α_{1}(z_{0}). In particular, uu^{∗} = z_{0}.

Moreover, by multiplying the relation U′_{g}u(t_{n}) = u(t_{n})U_{t_{n}}U′_{g}U_{t_{n}}^{∗}, ∀g ∈ G

(which holds true by 4.3.1^{∘}) from the right by α_{t_{n}}(u(t_{n})) , it follows that U′_{g}(u(t_{n})α_{t_{n}}(u(t_{n}))) = u(t_{n})U_{t_{n}}U′_{g}U_{t_{n}}^{∗}α_{t_{n}}(u(t_{n}))

= u(t_{n})U_{t_{n}}U′_{g}U_{t_{n}}^{∗}(U_{t_{n}}u(t_{n})U_{t_{n}}^{∗}) = u(t_{n})U_{t_{n}}(U′_{g}u(t_{n}))U_{t_{n}}^{∗}
= u(t_{n})U_{t_{n}}(u(t_{n})U_{t_{n}}U′_{g}U_{t_{n}}^{∗})U_{t_{n}}^{∗}
= (u(t_{n})α_{t_{n}}(u(t_{n})))(U_{t_{n}}^{2}U′_{g}U_{t_{n}}^{2∗})

Recursively, we obtain that u = n−1 ; ∏_{k=0}α_{t_{n}}^{k} satisfies U′_{g}u = u(U_{t_{n}}^{n})U′_{g}(U_{t_{n}}^{n})^{∗}, which by

taking into account that U_{t_{n}}^{n} = U_{1} gives U′_{g}u = uU_{1}U′_{g}U_{1}^{∗}.

Since U′_{g} = (w_{g} ⊗ 1)U_{g} and U_{1}(w_{g} ⊗ 1)U_{1}^{∗} = 1 ⊗ w_{g}, from this last relation we get (w_{g} ⊗ 1)U_{g}u = u(1 ⊗ w_{g})U_{g}, as desired. Q.E.D.

For the next lemma and the end of the proof of Theorem 4.1 we’ll use the notations in Proposition 3.4. Thus, we denote by H the Hilbert space of Hilbert-Schmidt operators on L^{2}(N p, φ), by σ^{w} the projective representation of G on L^{2}(N p, φ) given by ξ ↦ σ_{g}(ξ)w_{g}^{∗} and by σ̃^{w} the representation of G on H implemented by σ^{w} (see 3.4.2^{∘}).

4.7. Lemma. Assume the representation σ̃^{w} of G on H does not contain the trivial representation. Given any finite dimensional subspace H_{0} ⊂ L^{2}(N p, φ) and any ε > 0, there exists g ∈ G such that if P_{0} denotes the orthogonal projection of L^{2}(N p, φ) onto H_{0}, regarded as an element in H, then 0 ≤ T r(P_{0}σ̃_{g}^{w}(P_{0})) < ε.

Proof. This is quite standard, but we prove it in details for convenience. We denote

K(P_{0}) = \overline{co}^{w}\{σ̃_{g}^{w}(P_{0}) | g ∈ G\} ⊂ H.

Since K(P_{0}) is convex and weakly closed, by the inferior semicontinuity of the norm in H with respect to the weak topology, it follows that there exists an element Y_{0} ∈ K(P_{0}) with minimal Hilbert-Schmidt norm. Also, by the convexity of K(P_{0}), Y_{0} is the unique element with this property. Since σ̃_{g}^{w}(Y_{0}) belongs to K(P_{0}) and has the same norm in H as Y_{0}, it follows that σ̃_{g}^{w}(Y_{0}) = Y_{0}, ∀g ∈ G. Since σ̃^{w} does not contain the trivial representation, it follows that Y_{0} = 0.

But if for some ε > 0 we would have ReT r(P_{0}σ̃_{g}^{w}(P_{0})) ≥ ε, ∀g ∈ G, then by taking convex combinations and weak closure, we would have ReT r(P_{0}Y) ≥ ε, ∀Y ∈ K(P_{0}). Taking Y = Y_{0} = 0, this gives a contradiction. Thus, there exists g ∈ G such that ReT r(P_{0}σ̃_{g}^{w}(P_{0})) < ε.

Since P_{0} is a projection and σ̃_{g}^{w}(P_{0}) = σ^{w}(g)P_{0}σ^{w}(g)^{∗}, it follows that σ̃_{g}^{w}(P_{0}) is a projection as well. Thus, T r(P_{0}σ̃_{g}^{w}(P_{0})) ≥ 0. Q.E.D.

Proof of 4.1.1^{∘}. By 3.4.3^{∘}(v) there exists a projection q_{0} ∈ B_{0} such that q_{0}w is of the form (4.1.1) and such that there are no non-zero projections q_{1} ≤ (p − q_{0}), q_{1} ∈ B_{0} with q_{1}w of the form (4.1.1). Note right away that q_{0} ≤ z, where z is the maximal atomic projection of B_{0}. Indeed, this is because q_{0}w is a generalized weak 1-cocycle with Ad(q_{0}w) ∘ σ having atomic fixed point algebra (cf. 3.2.3^{∘}).

We have to prove that q_{0} = z, i.e., (z−q_{0}) = 0. We will proceed by contradiction, assuming that (z − q_{0}) ≠ 0. By replacing if necessary w by (z − q_{0})w, which is still a generalized weak 1-cocycle for σ, with the same scalar 2-cocycle as w, we may assume that the algebra B_{0} = (pNp)^{σ′} is atomic and, furthermore, that there exists no non-zero projection q_{1} ≤ p = 1_{B_{0}}, q_{1} ∈ B_{0}, such that q_{1}w is of the form (4.1.1).

By Proposition 3.4 the latter condition is equivalent to the fact that the representation σ̃^{w} of G on the Hilbert space H of Hilbert-Schmidt operators on L^{2}(N p, φ) does not contain the trivial representation.

But H can be identified with the space L^{2}(N p, φ)\overline{⊗}L^{2}(pN , φ). Under this identification, the representation σ̃^{w} becomes:

σ̃^{w}(g)(x′ ⊗ x′′^{∗}) = (σ_{g}(x′)w_{g}^{∗}) ⊗ (w_{g}σ_{g}(x′′)^{∗}), g ∈ G, x′, x′′ ∈ N p.

By Lemma 4.6 there exists a partial isometry u in Ñ with left support p = 1_{B_{0}} ∈ B_{0} = B_{0} ⊗ ℂ ⊂ N ⊗ ℂ and right support α_{1}(p) = 1 ⊗ p and satisfying:

(w_{g} ⊗ 1)U_{g}u = u(1 ⊗ w_{g})U_{g}, ∀g ∈ G.

Note that u actually belongs to (p ⊗ 1)Ñ(1 ⊗ p) ⊂ (pN)\overline{⊗}(N p). We want to prove that p = 0. To obtain this, we’ll prove that

(*)  φ(u(x ⊗ y^{∗})) = 0, ∀x, y ∈ N

Since the elements of the form x ⊗ y^{∗}, with x, y ∈ N, are total in N \overline{⊗}N, this will show that u ∈ Ñ is equal to zero, implying that p = uu^{∗} = 0. Since φ(u(x ⊗ y^{∗})) = φ(u((xq) ⊗ (yq)^{∗})), it is sufficient to establish (*) for x, y ∈ N p.

For each g ∈ G, by Lemma 4.6 we get:

φ(u(x ⊗ y^{∗})) = φ(U′_{g}(u(x ⊗ y^{∗}))U′_{g}^{∗})
= φ((w_{g} ⊗ 1)U_{g}(u(x ⊗ y^{∗}))U_{g}^{∗}(w_{g}^{∗} ⊗ 1))
= φ((u(1 ⊗ w_{g})U_{g}(x ⊗ y^{∗}))U_{g}^{∗}(w_{g}^{∗} ⊗ 1))
= φ((u(1 ⊗ w_{g})(σ_{g}(x) ⊗ σ_{g}(y^{∗}))(w_{g}^{∗} ⊗ 1))
= φ(u(σ_{g}(x)w_{g}^{∗} ⊗ w_{g}σ_{g}(y^{∗}))).

But if we identify the Hilbert spaces H and L^{2}(N p, φ)\overline{⊗}L^{2}(N p, φ)^{∗} in the usual way, then we have

(σ_{g}(x)w_{g}^{∗}) ⊗ (w_{g}σ_{g}(y^{∗})) = σ̃^{w}(g)(x ⊗ y^{∗})

so that by Lemma 4.7, for any given δ > 0 we can find recursively g_{1}, g_{2}, ..., g_{m} ∈ G such that π(g_{k+1})(x ⊗ y^{∗}) is δ orthogonal (with respect to the scalar product implemented by φ̃ = φ⊗φ) to each of the vectors π(g_{i})(x⊗y^{∗}), i = 1, 2, ..., k. Thus, the vectors {(1 ⊗ w_{g_{j}})U_{g_{j}}(x ⊗ y^{∗})U_{g_{j}}^{∗}(w_{g_{j}}^{∗} ⊗ 1)}_{1≤j≤m}, are mutually δ-orthogonal. By the Cauchy-Schwartz inequality we thus obtain:

|φ(u(x ⊗ y^{∗}))|^{2} = |φ(u1/m(Σ_{j}(1 ⊗ w_{g_{j}})U_{g_{j}}x ⊗ y^{∗})U_{g_{j}}^{∗}(w_{g_{j}}^{∗} ⊗ 1)))|^{2}

≤ ∥u∥_{2}^{2} ∥1/m(Σ_{j}(1 ⊗ w_{g_{j}})U_{g_{j}}(x ⊗ y^{∗})U_{g_{j}}^{∗}(w_{g_{j}}^{∗} ⊗ 1))∥2 ; φ̃

≤ ∥u∥_{2}^{2} ∥x ⊗ y∥2 ; φ̃/m + m(m − 1)δ.

Since δ > 0 was arbitrary, independently of m, taking first δ → 0 then m → ∞, we are done. Q.E.D.

In order to prove 4.1.2^{∘} (and thus end the proof of 4.1) we need to show that in

case (N_{0}, φ_{0}) is taken to be of the special form ⊗_{i=1}^{k}(N_{0,i}, φ_{0,i}), where N_{0,i} ≃ M_{2×2}(ℂ)

and φ_{0,i} are faithful states on N_{0,i}, 1 ≤ i ≤ k, then the fixed point algebra B_{0} must be atomic.

We do this in the next two lemmas.

4.8. Lemma. With the assumptions of 2.5.1, let α be as in the first part of 2.5.1 and β ∈ AutÑ be a period 2 automorphism like in the last part of 2.5.1, i.e., such that βα_{t}β^{−1} = α_{−t} and such that N ⊗ ℂ ⊂ Ñ^{β}. Assume there exists a diffuse weakly closed *-subalgebra B^{0} ⊂ N = N ⊗ ℂ and a partial isometry v_{0} ∈ Ñ such that v_{0}^{∗}v_{0} = 1_{B^{0}} and v_{0}B^{0}v_{0}^{∗} ⊂ α_{1/2^{n}}(N ⊗ ℂ), for some n ≥ 1. Then v_{0} = 0.

Proof. We will construct by induction over k some partial isometries v_{k} ∈ Ñ and diffuse weakly closed von Neumann subalgebras B_{k} ⊂ N ⊗ ℂ such that

(4.8.1)  τ(v_{k}v_{k}^{∗}) = τ(vv^{∗}), v_{k}^{∗}v_{k} = 1_{B_{k}}, v_{k}B_{k}v_{k}^{∗} ⊂ α_{1/2^{n−k}}(N ⊗ ℂ)

Letting B_{0} = B^{0}, v_{0} = v_{0}, we see that the relation holds true for k = 0. Assume we have constructed v_{j}, B_{j} for j = 0, 1, …, k. By applying the automorphism β of 2.5.1 to the inclusion in (4.8.1) and using the properties of β, it follows that

(4.8.2)  β(v_{k})B_{k}β(v_{k})^{∗} ⊂ α_{−1/2^{n−k}}(N ⊗ ℂ)

By further applying α_{1/2^{n−k}} to this latter inclusiuon it follows that

(4.8.3)  B_{k+1} =^{def} α_{1/2^{n−k}}(β(v_{k}))α_{1/2^{n−k}}(B_{k})α_{1/2^{n−k}}(β(v_{k}^{∗})) ⊂ N ⊗ ℂ

By conjugating (4.8.3) with α_{1/2^{n−k}}(β(v_{k}^{∗})) we thus get:

(4.8.4) α_{1/2^{n−k}}(β(v_{k}^{∗}))B_{k+1}α_{1/2^{n−k}}(β(v_{k})) = α_{1/2^{n−k}}(B_{k}) ⊂ α_{1/2^{n−k}}(N ⊗ ℂ)

On the other hand, by applying α_{1/2^{n−k}} to (4.8.1) we also have:

(4.8.5)  α_{1/2^{n−k}}(v_{k})α_{1/2^{n−k}}(B_{k})α_{1/2^{n−k}}(v_{k}^{∗}) ⊂ α_{1/2^{n−k−1}}(N ⊗ ℂ)

Altogether, it follows that if we let v_{k+1} = α_{1/2^{n−k}}(v_{k})α_{1/2^{n−k}}(β(v_{k}^{∗})) then by (4.8.4) and (4.8.5) we get:

v_{k+1}B_{k+1}v_{k+1} ⊂ α_{1/2^{n−k−1}}(N ⊗ ℂ)

Moreover, since β(v_{k}^{∗}v_{k}) = v_{k}^{∗}v_{k}, we also have v_{k+1}v_{k+1}^{∗} = α_{1/2^{n−k}}(v_{k}v_{k}^{∗}), so that τ(v_{k+1}v_{k+1}^{∗}) = τ(v_{k}v_{k}^{∗}).

This ends the induction argument. Taking k = n, it follows that v_{n}B_{n}v_{n} ⊂ α_{1}(N ⊗ ℂ) = ℂ ⊗ N, because α_{1} is the flip automorphism.

But by (Lemma 4.3 in [Po3], or 2.6 in [Po6]), this implies that v_{n} = 0. Since τ(v_{n}v_{n}^{∗}) = τ(vv^{∗}), it follows that v = 0. Q.E.D.

4.9. Corollary. Under the assumptions of 4.1.2^{∘}, the algebra B_{0} = N^{σ′} follows atomic.

Proof. By 4.3 and 4.4 it follows that if n ≥ 1 is sufficiently large then the partial isometry v′ = u(1/2^{n})^{∗} is non-zero, has right support in B_{0} and satisfies v′B_{0}v′^{∗} ⊂ α_{1/2^{n}}(N ⊗ ℂ). But if we assume B_{0}v′^{∗}v′ has a non-zero diffuse part B^{0} and we let v_{0} = v′1_{B^{0}}, then 4.8 applies to infer that v_{0} = 0, a contradiction. Thus, v′^{∗}v′B_{0}v′^{∗}v′ is atomic.

Since v′^{∗}v′ = u(1/2^{n})u(1/2^{n})^{∗} tends to p as n tends to ∞, it follows that B_{0} is atomic. Q.E.D. Proof of 4.1.2^{∘}. By Lemma 4.9 the algebra B_{0} follows atomic. But then, 4.1.1^{∘} applies to obtain that w is necessarily of the form (3.2.1). Q.E.D.

5. Some consequences and comments.

The first application of 4.1 that we are going to emphasize is that, for an action of the property T group G by Connes-Størmer Bernoulli shifts σ on N = N_{φ}, the multiplicative group H ⊂ ℝ_{+}^{∗} arising as the spectrum of the discrete decomposition of the associated ITPF1 factor (N , φ) is an outer conjugacy invariant for σ. In fact, we define a certain outer conjugacy invariant for arbitrary actions σ of groups G, and then use 4.1.1^{∘} to prove that if the group G has the property T then this invariant can be explicitly calculated, being equal to H_{1} = {β ∈ H | β ≤ 1}.

5.1. Definition. Let σ be an action of a group G on a type II_{1} factor N. Let H_{σ}^{1} be the set of generalized weak 1-cocycles w of σ satisfying the following properties:

1^{∘}. The fixed point algebra of the action Ad(w_{g}) ∘ σ_{g}, g ∈ G, on pNp is atomic, where p is the support of w.

2^{∘}. There exists an atomic von Neumann subalgebra B_{σ}^{w} ⊂ pNp such that Ad(w_{g}) ∘ σ_{g}(B_{σ}^{w}) = B_{σ}^{w}, ∀g ∈ G and such that if B ⊂ pNp is atomic and Ad(w_{g}) ∘ σ_{g}(B) = B, ∀g ∈ G then B ⊂ B_{σ}^{w}.

For each w ∈ H_{σ}^{1} let P_{σ}^{w} be the set of minimal (non-zero) projections of B_{σ}^{w}. Define

H(σ) = {τ(q) | ∃w ∈ H_{σ}^{1}, q ∈ P_{σ}^{w}},

with the convention that H(σ) = {0} if H_{σ}^{1} = ∅

5.2. Proposition. H(σ) is an outer conjugacy invariant for σ.

Proof. By the definitions, it follows that if w is a weak 1-cocycle for σ then H_{σ}^{1} = H_{σ′}^{1}w^{∗}, where σ′ = Ad(w) ∘ σ. Moreover, if w′ ∈ H_{σ}^{1} then B_{σ}^{w′} = B_{σ′}^{w′w^{∗}}. This implies H(σ) = H(σ′). Q.E.D.

5.3. Theorem. 2^{∘}. Assume G is an infinite group with the property T of Kazhdan and σ is an action of G on the hyperfinite type II_{1} factor by Connes-Størmer Bernoulli shifts, as in the hypothesis of 4.1. Let H_{1} ⊂ (0, 1] be the set associated with the discrete decomposition of the ITPF1 factor (N , φ) on which σ is defined, as in 2.2.0 − 2.2.2 (thus, H_{1} = {β ∈ H | β ≤ 1}, with H being the spectrum of the discrete decomposition of (N , φ)). Then H(σ) = H_{1}.

Moreover, if N is of type III (equivalently, H ≠ {1}), 0 < t ≤ 1 and σ^{t} is chosen to be a genuine action (cf. 3.3), then H(σ^{t}) = {β/t | β ∈ H_{1}, β ≤ t}.

Proof. To prove the first part, note that by Theorem 4.1.1^{∘}, any generalized weak 1cocycle w ∈ H_{σ}^{1} is of the form (3.2.1). But for such cocycles, Proposition 3.4 shows that there exists indeed an atomic von Neumann subalgebra invariant to Adw ∘ σ satisfying the maximality condition 5.1.2^{∘}. Moreover, by 3.2, 3.4 the traces of the minimal projections in this atomic algebra are in the set H_{1}. Thus, H(σ) ⊂ H_{1}.

Conversely, if β_{0} ∈ H_{1} then there always exist some non-negative integers n_{β}, β ∈ H_{1}, such that n_{β_{0}} ≠ 0 and Σ_{β}n_{β}β = 1 (exercise !). For each β ∈ H_{1} take π_{β} to be a n_{β}-multiple of the trivial representation of G, on the Hilbert space ℓ^{2}(n_{β}). Let w be a 1-cocycle constructed out of {π_{β}}_{β}, for some appropriate choice of isometries in V, as in (3.2.1). By 3.4 it follows that w ∈ H_{σ}^{1} and by 3.2 the corresponding B_{σ}^{w} is given by the algebra B constructed in 3.2.2^{∘}. By construction, it follows that if q is a minimal projection in B_{σ}^{w} then τ(q) = β_{0}. Thus, H_{1} ⊂ H(σ) as well.

To prove the last part of the statement, let w be a generalized 1-cocycle for σ of support p, with τ(p) = t such that σ^{t} = Ad(w) ∘ σ. Note that any 1-cocycle w′ for σ^{t} gives rise to a generalized 1-cocycle w′w for σ. Thus, by 4.1.1^{∘}, w′w is of the form (3.2.1), having an atomic invariant subalgebra B described by 3.2.2^{∘}. Moreover, by 3.2.3^{∘}, B contains all other atomic subalgebras of N invariant to σ^{t}. Then 3.2.3^{∘} applies to show that the traces of the minimal projections on B are of the form β/t with β ∈ H_{1}, β ≤ t. The other inclusion is proved in the same way as the similar inclusion above. Q.E.D.

5.4. Corollary. Let σ, σ′ be two Connes-Størmer Bernoulli shifts of the property T group G on the hyperfinite type II_{1} factor. Let H, H′ be the corresponding multiplicative groups, arising as the spectra of the discrete decompositions of the ITPF1 factors (N , φ), (N′, φ′) on which σ, σ′ are defined. Let t, t′ > 0 and σ^{t}, σ^{t′} be cocycle actions defined as in 3.1.1, out of σ, σ′. If σ^{t} is outer conjugate to σ′^{t′} then H = H′ and t/t′ ∈ H. If in addition σ^{t}, σ′^{t′} are genuine actions, then they follow conjugate.

Proof. Let s = max{t, t′}. Then we have (σ^{t})^{1/s} is outer conjugate to (σ′^{t′})^{1/s}. This shows that we may assume t′ = 1 and t ≤ 1.

There are two possibilities: Either N is of type II_{1} and H = {1} or N is of type III.

If N = N is of type II_{1} and t < 1 then σ^{t} outer conjugate to σ′ implies there exists a generalized 1-cocycle w for σ of support p such that τ(p) = t and such that Ad(w) ∘ σ is conjugate to σ′. In particular, Ad(w) ∘ σ is ergodic, so it has trivial (in particular atomic) fixed point algebra. By 4.1.1^{∘} it follows that w is given by (3.2.1). But H = {1}, so p is forced to be equal to 1 and w to be a weakly trivial weak 1-cocycle. Thus t = 1 and σ follows conjugate to σ′. In particular, by 4.1.1^{∘}, H = H = {1}.

If N is of type III, then by 3.3 we can choose σ^{t} to be a genuine action. Since 1 ∈ H′_{1} and H′_{1} = H(σ′) (cf. 5.3) it follows that 1 ∈ H(σ′). But by 5.3, we also have H(σ^{t}) = {β/t | β ∈ N, β ≤ t}. Since σ is outer conjugate to σ^{t} and H(σ) is an outer conjugacy invariant, we have 1 ∈ H(σ^{t}). The only situation when 1 can be contained in H(σ^{t}) is when t belongs to H_{1} = H(σ). But then (N^{t}, σ^{t}) is conjugate to (N, σ) by Adv, where v is an isometry in V_{t}. Thus, σ follows outer conjugate to σ′, implying that H(σ) = H(σ′).

Moreover, if σ′ (which is strongly mixing) is conjugate to Ad(w) ∘ σ for some weak 1-cocycle w for σ, then the fixed point algebra of Ad(w) ∘ σ is equal to ℂ1 and by 4.1.1^{∘} it follows that w is of the form (3.2.1). Also, since Ad(w) ∘ σ follows strongly mixing itself (as being conjugate to σ′), it leaves no non-trivial atomic von Neumann subalgebra invariant. By 4.1.1^{∘}, this implies the weak 1-cocycle w is weakly trivial. Thus, σ′ ∼ σ. Q.E.D.

We now derive some consequences of Theorem 4.1 and of the above results, in the case of “classical” Bernoulli shifts. The first such consequence is derived from 4.1.1^{∘}, being a particular case of 5.4 and dealing with arbitrary Bernoulli shifts. The other two Corollaries are derived from 4.1.2^{∘} and they are proved for M_{2^{n}×2^{n}}(ℂ)-Bernoulli shifts only.

5.5. Corollary. Let σ_{i} be M_{n_{i}×n_{i}}(ℂ)- Bernoulli shifts, i = 1, 2, of the infinite property T group G on the hyperfinite type II_{1} factor, as in 2.1. Let also t_{1}, t_{2} > 0. If σ_{1}^{t_{1}} is outer conjugate to σ_{2}^{t_{2}} then t_{1} = t_{2} and σ_{1} is conjugate to σ_{2}. In particular, taking n_{1} = n_{2} = n, σ_{1} = σ_{2} = σ, t_{1} = 1 and t_{2} = m it follows that the genuine actions {σ ⊗ id_{m}}_{m} are mutually non-outer conjugate.

Proof. This is just a particular case of 5.6. Q.E.D.

5.6. Corollary. Let σ be a M_{2^{n}×2^{n}}(ℂ)-Bernoulli shift action of the infinite property T group G on the hyperfinite type II_{1} factor N. Then any weak 1-cocycle w for σ is weakly trivial, i.e., if w : G → U(N) is so that Ad(w_{g}) ∘ σ_{g} is an action of G on N, then there exists a unitary element v ∈ U(N) such that w_{g} = v^{∗}σ_{g}(v), modℂ, ∀g ∈ G. Moreover, the set of 1-cocycles for σ, modulo equivalence, coincides with the set of characters of G, thus being finite.

Proof. By 4.1.2^{∘}, since in the case of classical Bernoulli shifts the multiplicative group H is equal to {1} and V = U(N), it follows that any weak 1-cocycle w with scalar 2-cocycle μ is of the form w_{g} = λ_{g}v^{∗}σ_{g}(v) for some unitary element v ∈ N and projective one dimensional representation λ : G → 𝕋 having scalar 2-cocycle μ. That implies λ_{e} = 1 and μ_{g,h} = λ_{g}λ_{h}\overline{λ_{gh}}, ∀g, h ∈ G.

In particular, if w is a 1-cocycle then μ = 1 and λ follows a character. Moreover, if λ is a character and v is a unitary element in N such that λ_{g} = v^{∗}σ_{g}(v), ∀g ∈ G then σ_{g}(v) ∈ ℂv, ∀g. Since σ is strongly mixing, by 2.4.3 this implies v ∈ ℂ itself. Thus, different characters give different 1-cocycles. The number of characters is finite by ([K]). In fact, by ([V2]), even the number of equivalence classes of representations of a fixed dimension of a group with the property T is finite. Q.E.D.

5.7. Corollary. Assume σ, G are as in 5.6. For each 1 > t > 0, let σ^{t} be a choice of a cocycle action on N^{t} ≃ R obtained by reducing, or amplifying the action σ, as in 3.1.1. Then σ^{t} cannot be perturbed by inner automorphisms to a genuine action.

Proof. If σ^{t} would be perturbable to a genuine action then σ would have a generalized weak 1-cocycle w of support p, with p ∈ N a projection of trace t < 1. But in the case of classical Bernoulli shifts the group H is equal to {1} and by 4.1.2^{∘}, w must be of the form (3.2.1), thus it must have support equal to 1. Q.E.D.

Let us point out that even milder rigidity properties on a group G can be suficient to ensure that results similar to 5.5-5.7 hold true:

5.8. Theorem. All the conclusions in 5.5 − 5.7 hold true for the group G = SL(2, ℤ) ⋊ ℤ^{2} as well.

Proof. The only modifications needed, from the proof of the case G has the property T, are as follows:

Assume w is a generalized 1-cocycle for the M_{2^{k}×2^{k}}(ℂ)-Bernoulli shift σ of the group G = SL(2, ℤ) ⋊ ℤ^{2} on N = \overline{⊗}_{g}(M_{2^{k}×2^{k}}(ℂ), tr)_{g}. Then in Lemma 4.2 one defines the set K′_{t}(x) as the weak closure of the convex hull of elements of the form U′_{g}xU_{t}U′_{g}^{∗}U_{t}^{∗}, with g running over the subgroup ℤ^{2} (rather than over all G). The proofs of 4.2, 4.3 do work in this case as well (cf. observation before statement of lemma 4.2).

Next, Lemma 4.4 holds true, due the rigidity of the inclusion ℤ^{2} ⊂ SL(2, ℤ)⋊ℤ^{2} (cf. [K], [DKi]). All the proofs of 4.5-4.7 (+ the proof of 4.1.1^{∘}) are the same, to conclude that if z denotes the maximal atomic projection in B_{0} then the restriction of zw to ℤ^{2} is a weak coboundary, if nozero, i.e., for some apropriate v ∈ R one has zw_{g} = v^{∗}σ_{g}(v), modℂ, ∀g ∈ ℤ^{2}. This already implies that the support of zw is equal to 1, due to the ergodicity of the action σ_{|ℤ^{2}} on N (this restriction being a R-Bernoulli shift).

Moreover, by replacing w_{g} by w′_{g} = vw_{g}σ_{g}(v^{∗}), g ∈ G one gets a weak cocycle for σ which is scalar valued when restricted to ℤ^{2}. Let a, b be the generators of SL(2, ℤ). It follows that Adw′_{a}σ_{a} and Adw′_{b}σ_{b} normalize σ(ℤ^{2}), implying that w′_{a}σ_{g}(w′_{a}^{∗}) and w′_{b}σ_{g}(w′_{b}^{∗}) are scalars ∀g ∈ ℤ^{2}. Thus, σ_{g}(w′_{a}) ∈ ℂw′_{a} and similarily for b. Since the only finite dimensional vector subspace of L^{2}(N) that is invariant to σ_{|ℤ^{2}} is ℂ1 (cf. 2.4.3), it follows that w′_{a}, w′_{b} are scalars. Thus, w_{g} = v^{∗}σ_{g}(v), modℂ, ∀g ∈ G.

This proves 4.1.1^{∘} in the case of classical M_{n×n}(ℂ)- Bernoulli shifts. But then 4.1.2^{∘} follows by simply noticing that Lemmas 4.8 did not require G to have the property T and deducing 4.9 from this lemma and the version of 4.4.3^{∘}3, 4.5 that are valid for G = SL(2, ℤ) ⋊ ℤ^{2} (as explained bove). Q.E.D.

5.9. Remarks. 1^{∘}. The results obtained in 4.1.2^{∘}, 5.6.2^{∘}, 5.7 and (part of) 5.8 are for (N_{0}, φ_{0})- Connes-Størmer Bernoulli shifts, in which (N_{0}, φ_{0}) is a tensor product of 2 by 2 matrix algebras with faithful states on them. The only time we need this special type of algebra (N_{0}, φ_{0}) is for the construction of the period 2 automorphism β in Lemma 2.5.1. This automorphism, in turn, is being used only in the proof of Lemma 4.8. In fact, one can avoid using the existence of β in the proof of 4.8.

More precisely, one can prove 4.8 for (N_{0}, φ_{0})-Connes-Størmer Bernoulli shifts with (N_{0}, φ_{0}) = (M_{k×k}(ℂ), φ_{0}) for arbitrary k and arbitrary faithful state φ_{0}. Unfortunately though, the argument we have is rather long and complicated, due to the more involved geometry of the corresponding unitary groups U(2k), especially when taking into account the centralizer of φ_{0} ⊗ φ_{0} in this group. With this, however, the conclusions in the above mentioned results follow for all Connes-Størmer Bernoulli shifts of a property T group G or for G = SL(2, ℤ) ⋊ ℤ^{2}.

5.9.2^{∘}. We only needed the assumption that σ is a Bernoulli shift for the ergodicity properties that such actions have (strongly mixing) and for the rich structure of the commutant of σ in AutN. Thus, the following assumptions on an action σ on a type II_{1} factor N are in fact sufficient to insure that a statement similar to 4.1 holds true:

(a). G is an infinite property T group.

(b). N is a type II_{1} factor and (N , φ) is a von Neumann factor with N_{φ} = N, such that (N , φ) has a discrete decomposition over N.

(c). σ is a strongly mixing action of G on (N , φ).

(d). There exists a type II_{1} factor Ñ containing N such that σ extends to an action θ of G on Ñ and such that there exists a one parameter group of automorphisms α of ℝ on Ñ, commuting with the action θ and with the property that if there exists a partial isometry u ∈ Ñ satisfying w_{g}θ_{g}(u) = uα_{1}(w_{g}), ∀g ∈ G, for some generalized weak 1-cocycle w for σ then w is of the form (3.2.1).

(e). There exists a period 2 automorphism β of Ñ such that N ⊂ Ñ^{β}, βα_{t}β = α_{−t}, ∀t.

(f). If a partial isometry u ∈ Ñ satisfies u^{∗}u = 1_{B^{0}}, uB^{0}u^{∗} ⊂ α_{1}(B^{0}) for some diffuse von Neumann algebra B^{0} ⊂ N then u = 0.

Thus, conditions (a)-(d) imply 4.1.1^{∘} and together with the conditions (e)-(f) they imply 4.1.2^{∘} as well. For Bernoulli shifts in Sections 4 and 5 we have taken Ñ = N\overline{⊗}N and θ = σ ⊗ σ. In Section 6 we will give another example when this abstract framework can be realized.

5.10. Remarks. Let us mention some more remarks related to the vanishing cohomology results in this section, as well as some problems:

5.10.1. Note that Corollary 5.5 cannot distinguish between the M_{n×n}(ℂ)- Bernoulli shifts, for n = 2, 3, .... This points towards the following problem:

Let G be an infinite group and σ_{i} the M_{n_{i}×n_{i}}(ℂ)-Bernoulli shifts, i = 1, 2, of the group G. Is it true that σ_{1} conjugate to σ_{2} implies n_{1} = n_{2} ?

If solved in the affirmative then, together with the Corollary 5.5, the M_{n×n}(ℂ)- Bernoulli shifts by a property T group G would follow mutually non-outer conjugate, n ≥ 2.

Note that the case G is amenable of this problem was proved in the affirmative by Kawahigashi in ([Ka]). But the invariant he used, which is an addaption to the case of actions by amenable groups of the Connes-Størmer entropy, doesn’t work beyond that case, at least not in an evident way. Note in this respect that if an entropy invariant could be defined for non-amenable groups as well, along the lines of ([CS, Ka]), then by using 5.3 one could easily find Connes-Størmer Bernoulli shifts σ with distinct invariant H(σ) yet same entropy.

5.10.2. One can also prove a (slightly modified) version of 3.2, 3.4 and 4.1 for actions σ ⊗ id_{n} on N ⊗ M_{m×m}(ℂ) with σ as in 4.1. As a result, one obtains a calculation of the invariant H(σ ⊗ id_{m}) for these actions. As a consequence, one gets some versions of 5.2.4^{∘}, 5.5.1^{∘} for all t > 0 (not just for 1 ≥ t > 0).

5.10.3. Any finitely generated subgroup H of ℝ_{+}^{∗} can be realized as the invariant H(σ) of some Connes-Størmer Bernoulli shift of the type considered in 5.3.2^{∘}.

5.10.4. Note that M. Choda has already constructed in ([Ch]) a continuous family of mutually non-conjugate actions by the property T groups G = SL(n, ℤ) on R, for n ≥ 3. Moreover, these actions were proved to be mutually non-cocycle conjugate, modulo countable sets, in ([Ka]). The argument in ([Ka]) relies on first proving that the set of ergodic 1-cocycles of an ergodic action σ by a property T group G is countable, then using ([Ch]). Note that in this respect, our Corollary 5.6 gives a list of all 1-cocycles (not just the “ergodic” ones), in the special case when σ is a M_{2^{n}×2^{n}}(ℂ)-Bernoulli shift, but for arbitrary property T groups G.

Note that the actions of G = SL(n, ℤ) considered in ([Ch, Ka]) are not Bernoulli shifts. Indeed, those actions are so that the corresponding cross products R ⋊ G have the property T as von Neumann algebras (in the sense of [CJ]), while the cross product algebras associated to Bernoulli shift actions can never have the property T (cf. [Ch]). This is because the automorphism group of such a cross product is large (for instance, AutR/IntR contains a copy of 𝕋, see 2.5.1).

5.10.5. It seems to us that some of the results 4.1, 5.2-5.5 could be true for a larger class of non-amenable groups, than the property T and SL(2, ℤ) ⋊ ℤ^{2} we considered here. Related to this, the following question is of obvious interest:

What is the class 𝔾(R) of all groups G for which any cocycle action on the hyperfinite type II_{1} factor R can be perturbed to a genuine action ?

Note that besides the amenable groups (cf. [Oc]), the class 𝔾(R) of such groups contains the free groups F_{n} (which have no 2-cocycles) and more generally any group G of the form G = G_{1} ∗_{H} G_{2} ∗_{H} G_{3}..., where G_{i} are amenable groups and H ⊂ G_{i} is a common finite group (cf. [Su, J1]). We will prove in another paper that this class contains some other products with amalgamation, such as 𝔽_{2} ⊕ ℤ = ℤ^{2} ∗_{ℤ} ℤ^{2}. Note that not all free products with amalgamation of amenable groups belongs to 𝔾(R). Indeed, by 5.8 we have that G = SL(2, ℤ) ⋊ ℤ^{2} does not belong to 𝔾(R).

6. Cohomology of free Bernoulli shifts.

We prove in this section that the same results as in 5.5-5.8 hold true for actions of infinite property T groups G (or of the group G = SL(2, ℤ) ⋊ ℤ^{2}) on the free group factor L(𝔽_{G}) ≃ L(𝔽_{∞}), by free Bernoulli shifts.

Thus, we consider a special case of the construction in 2.3, which for simplicity we describe in an alternative way: We lable the free generators of the group 𝔽_{∞} by {a_{g} | g ∈ G} and re-denote this group by 𝔽_{G}. We denote by σ the (left) action of the group G on L(𝔽_{G}) determined by:

σ_{g}(a_{h}) = a_{g^{−1}h}, ∀g, h ∈ G,

and call it the action of G on L(𝔽_{G}) by (left) free shifts.

6.1 Theorem. With G, L(𝔽_{G}), σ, as above, we have:

1^{∘}. Any weak 1-cocycle w for σ is weakly trivial and any 1-cocycle is equivalent to a character.

2^{∘}. For each t > 0, let σ^{t} be a choice of a cocycle action on (L(𝔽_{G}))^{t}(≃ L(𝔽_{G}), by [V3, Ra1]), obtained by reducing, or amplifying the action σ, as in 3.1. If 0 < t < 1 then σ^{t} cannot be perturbed by inner automorphisms to a genuine action.

3^{∘}. With the notation in 2^{∘}, the cocycle actions {σ^{t}}_{t>0}, all regarded as acting on L(𝔽_{G}) ≃ (L(𝔽_{G}))^{t}, are mutually non-outer conjugate. In particular, by taking t to be integers, it follows that the sequence of genuine actions {σ ⊗ id_{n}}_{n} on L(𝔽_{G}) are not outer conjugate.

Proof. For simplicity, denote N = L(𝔽_{∞}). To prove the statement it is suficient to show that if w is a generalized weak 1-cocycle with support p for the action σ then p = 1 and there exists v ∈ U(N) such that vw_{g}σ_{g}(v)^{∗} ∈ ℂ, ∀g ∈ G. Indeed, because then the arguments in the proof of 5.5 − 5.8 apply identically to get the above 1^{∘}, 2^{∘}.

The proof of the vanishing of the generalized weak 1-cocycles is basically the same as the proof of 4.1, except that the role of the tensor product N ⊗ N will this time be played by the free product N ∗ N. Due to this, it is convenient to keep the notation {a_{g}}_{g} for the generators of N ∗ ℂ and denote these same generators of ℂ ∗ N by {b_{g}}_{g}.

We split the proof in a number of “steps”.

Step 1. We first show that there exists a continuous action α : ℝ → Aut(N ∗ N) and a period 2 automorphism β ∈ Aut(N ∗ N) such that

(a). α_{1}(N ∗ ℂ) = N_{0} where N_{0} is free with respect to N = N ∗ ℂ and N, N_{0} generate N ∗ N.

(b). N ∗ ℂ ⊂ (N ∗ N)^{β}.

(c). βα_{t}β = α_{−t}, ∀t.

(d). α and β commute with the free shift θ = σ ∗ σ.

This construction plays the role of Lemma 2.5.1 in the sequel.

To construct these automorphisms, let h_{g} ∈ ℂ ∗ N be self-adjoint elements with spectrum in [0, 2] such that b_{g} = exp(πih_{g}), ∀g. We then put α_{t}(a_{g}) = exp(πith_{g})a_{g} and α_{t}(b_{g}) = b_{g}, ∀g ∈ G, t ∈ ℝ. By ([V1]) exp(πith_{g})a_{g} and b_{g} are mutually free and they clearly generate the same von Neumann algebra as a_{g}, b_{g}. Thus, α_{t} defines an automorphism of N, ∀t. Similarily, b_{g}a_{g} is free with respect to b_{g} and they jointly generate the same algebra as a_{g}, b_{g} do. Thus, (a) is satisfied.

Moreover, by the definition we clearly have α_{t}α_{s} = α_{t+s}, ∀t, s ∈ ℝ. Thus, α is a continuous action and it is immediate to see that it commutes with θ = σ ∗ σ.

Also, define β(a_{g}) = a_{g} and β(b_{g}) = b_{g}^{∗}, ∀g ∈ G. This clearly defines a period 2 automorphism of N ∗ N that commutes with σ.

Moreover, α_{1}(N ∗ ℂ) = N_{0}, with N, N_{0} generating N ∗ N and N ∗ ℂ ⊂ (N ∗ N)^{β}. Further on, we have

β(α_{t}(β(a_{g}))) = β(α_{t}(a_{g}))
= β(exp(πith_{g})a_{g}) = exp(πith_{g})^{∗}a_{g}
= exp(−πith_{g})a_{g} = α_{−t}(a_{g}).

Similarily, we get

β(α_{t}(β(b_{g}))) = b_{g} = α_{−t}(b_{g}).

This shows that all the required conditions are satisfied.

Step 2. Denote by σ̃^{w} the representation of G on L^{2}(Np, τ)\overline{⊗}L^{2}(pN, τ), as in 3.4. Apply 3.4.3^{∘} for N = N and H_{1} = {1} to deduce that if σ̃^{w} contains the trivial representation then p = 1 and there exists a unitary element v ∈ N such that vw_{g}σ_{g}(v)^{∗} ∈ ℂ, ∀g ∈ G. Thus, we are left to prove the Theorem 6.1 when σ̃^{w} doesn’t contain the trivial representation. More precisely, we want to show that this assumption leads to a contradiction.

Step 3. Denote σ′ = Ad(w) ∘ σ and B_{0} = (pNp)^{σ′}. Consider the action of G ⊕ ℝ implemented by the commuting actions θ = σ ∗ σ and α on N ∗ N. Then define p_{t} ∈ N ∗ N the same way as in Lemma 4.2 and u(t) the same way as in Lemma 4.3. Thus, u(t) is an isometry in N ∗ N such that if we put l(t) = u(t)u(t)^{∗}, r(t) = u(t)^{∗}u(t) then conditions 1^{∘}-3^{∘} of Lemma 4.3 are satisfied, where B_{0} is identified with B_{0} ∗ ℂ.

Step 4. The function t ↦ p_{t} follows continuous, with respect to the norm-2 topology, in exactly the same way as in Lemma 4.4, by using the property T of the group G.

Step 5. Lemma 4.8 holds in this “free product” context as well: Indeed, we just replace Ñ by N ∗ N and N ⊗ ℂ by N ∗ ℂ throughout the proof of that lemma to deduce that if there exists a diffuse von Neumann subalgebra B^{0} ⊂ N ∗ ℂ and a partial isometry v_{0} ∈ N ∗ N satisfying v_{0}^{∗}v_{0} = 1_{B^{0}} and v_{0}B^{0}v_{0}^{∗} ⊂ α_{1/2^{n}}(N ∗ ℂ) for some n ≥ 1 then there exists a partial isometry v_{n} ∈ N ∗ N and a diffuse algebra B_{n} ⊂ N ∗ ℂ such that τ(v_{n}v_{n}^{∗}) = τ(v_{0}v_{0}^{∗}), v_{n}^{∗}v_{n} = 1_{B_{n}} and v_{n}B_{n}v_{n}^{∗} ⊂ N_{0}, where N_{0} = α_{1}(N ∗ ℂ) as denoted in Step 1.

Since N = N ∗ ℂ and N_{0} are mutually free and they generate N ∗ N, by (4.3 in [Po6]) it follows that v_{n} = 0, thus v_{0} = 0 and B^{0} = 0.

Step 6. One applies the results in Steps 4, 5 to B_{0}, u(1/2^{n}) the same way as in the proof of Lemma 4.5, to first conclude that B_{0}l(1/2^{n}) is atomic, then pursue with the same argument as in that proof to conclude that all B_{0} follows atomic, and that on each minimal central projection of B_{0} Ad(u(1/2^{n})) implements α_{1/2^{n}}, for n sufficiently large.

Step 7. We conclude, by using exactly the same construction as in the proof of Lemma 4.6, that there exists a partial isometry u ∈ N ∗ N such that uu^{∗} = 1_{B_{0}} and

w_{g}U_{g}u = uα_{1}(w_{g})U_{g}, ∀g ∈ G,

where U_{g} are the canonical unitaries in (N ∗ N) ⋊ G implementing the action θ of G on N ∗ N.

Step 8. To end the proof of Theorem 6.1 we will prove that the relation in Step 7 entails u = 0, giving us the desired contradiction. To show this we prove that τ(uX) = 0, ∀X ∈ N ∗ N.

We use the fact that N ∗ N is generated by N = N ∗ ℂ and N_{0} = α_{1}(N ∗ ℂ), with N, N_{0} mutually free. We denote by b′_{g} = α_{1}(a_{g}). Thus, it is suficient to prove that τ(uX) = 0 for X of the form X_{1}Y_{1}X_{2}Y_{2}... or of the form Y_{0}X_{1}Y_{1}..., where X_{i} are words in a_{g} and Y_{j} are words in b′_{g}.

By the relation in Step 7 we have:

τ(uX) = τ(puXp) = τ(U′_{g}(uX)U′_{g}^{∗})
= τ(w_{g}U_{g}(uX)U_{g}^{∗}w_{g}^{∗}) = τ(uα_{1}(w_{g})U_{g}XU_{g}^{∗}w_{g}^{∗})
= τ(uα_{1}(w_{g})θ_{g}(X)w_{g}^{∗})

Thus, if X is a word of length at least 3 in X_{i}, Y_{i}, with all “letters” X_{i}, Y_{i} of trace 0, then the “middle” letters have only θ_{g} acting on them, being un-altered by the left multiplication by w_{g} ∈ N ∗ ℂ and by the right multiplication by α_{1}(w_{g}) ∈ N_{0}. Since u can be approximated arbitrarily well by an element in the algebra generated by {a_{h}, b′_{h} | g, h ∈ S}, for some finite subset S ⊂ G, it follows that we have

lim_{g→∞}τ(uα_{1}(w_{g})θ_{g}(X)w_{g}^{∗}) = 0

for any such X. From the above equations, this implies τ(uX) = 0. A similar argument shows that if X is of the form X_{1}Y_{1} with τ(X_{1}) = τ(Y_{1}) = 0 then τ(uX) = 0.

Thus, we are left to check the equality τ(uX) = 0 for X = Y_{0}X_{1}. But this brings us to a situation equivalent to the hypothesis of Lemma 4.7. By using Step 2 and Lemma 4.7 in the same way as in the Proof of 4.1.1^{∘}, it follows that τ(uX) = 0 in this case as well. Q.E.D.

6.2. Remark. Note that Theorem 6.1 provides a different class of cocycle actions by property T groups G on L(𝔽_{∞}) that cannot be perturbed to genuine actions, than the cocycle action of Connes and Jones ([CJ]; see the Appendix 2 in this paper).

Indeed, each of our cocycle actions σ^{t} on the factor (L(𝔽_{∞}))^{t} (which is isomorphic to L(𝔽_{∞}), ∀t, by [V3, Ra1]), obtained by reducing a free Bernoulli shift σ by a projection of trace t, has the property that its t^{−1} amplification can be perturbed to an action. While none of the amplifications of the cocycle action in ([CJ]) can be perturbed to an action. Indeed, the argument is the same: (L(𝔽_{n}))^{s} still has Haagerup’s compact approximation property ([H]), ∀s > 0, so L_{μ}(G) cannot be embedded in it, ∀μ a scalar 2-cocycle of G.

6.3. Remarks. 1^{∘}. Related to 6.1, to the examples in ([CJ]) and to the problem 5.7.3, the following question is in place:

Find the class 𝔾 of all groups for which any cocycle action on any type II_{1} factor can be perturbed to a genuine action.

Note that, like for cocycle actions on the hyperfinite type II_{1} factor, the class 𝔾 contains all amenable groups ([Po1]) and, more generally, all the free products of amenable groups with amalgamation over finite subgroups (this follows trivialy by applying [Su, J1] and [Po1]). We do not know of any other example of group in this class.

2^{∘}. The Connes-Jones cocycles seem to be “universally bad”, so the class of groups G (with some specified presentation 𝔽_{n} → G → 1) for which such cocycles vanish (or do not vanish) is an interesting test example to study for the Problem 6.3.1^{∘} above. The obvious obstruction for vanishing cohomology to look at in this case would be the non-embeddability of the group algebra L(G) into L(𝔽_{n}), like in the argument in ([CJ]). So the following problem (see [Ge]), seems to be very interesting :

What is the class of von Neumann algebras that can be embedded in the free group factors L(𝔽_{n}) ? In particular, what are the groups G for which L(G) can be embedded in L(𝔽_{n}) ?.

3^{∘}. Along these lines, Liming Ge has speculated that L(𝔽_{n}) does not contain noninjective von Neumann subalgebras N_{0} with N′_{0} ∩ L(F_{n}) diffuse (without atoms). Equivalently, if N_{0} is not injective and Z(N_{0}) ≃ L^{∞}([0, 1]) then N_{0} cannot be embedded in L(𝔽_{n}), n ≥ 2. If true, Ge’s conjecture would imply that a group G_{0} is non-amenable if and only if L(G_{0} ⊕ ℤ) cannot be embedded into L(𝔽_{n}). Related to this, Shlyakhtenko has asked whether the factor (L(𝔽_{∞})\overline{⊗}L^{∞}([0, 1], λ)) ∗ L(𝔽_{∞}) is isomorphic to L(𝔽_{∞}). An affirmative answer to Ge’s conjecture would imply these factors are not isomorphic.

In fact, it even seems possible that no thin type II_{1} factors ([Po2]) other than the hyperfinite ones can be embedded into L(𝔽_{n}). In particular, this would imply that no non-injective factors with the property Γ of Murray and von Neumann ([MvN]) can be embedded into L(𝔽_{n}) (such factors being thin by [GePo]). In this respect, let us point out the following:

6.4. Proposition. Assume the type II_{1} factor N (with separable predual) contains a non-injective von Neumann subalgebra N_{0} such that N′_{0} ∩ N^{ω} is a diffuse von Neumann algebra, where N^{ω} is an ultrapower algebra of N. Then there exists a non-injective von Neumann subalgebra N_{1} ⊂ N such that N′_{1} ∩ N is diffuse.

Moreover, if N_{0} is a factor then N_{1} can be taken to be a factor. In particular, if N contains a non-hyperfinite subfactor with the property Γ then it contains a non-hyperfinite subfactor with the property Γ with diffuse relative commutant.

Proof. Since N_{0} in not injective, there exists a finite set {x_{1}, x_{2}, …, x_{n}} ⊂ N_{0} such that the von Neumann algebra it generates is non-injective. By Connes’ characterisation of injectivity with hypertraces it follows that there exists ε > 0 such that if {x′_{1}, …, x′_{n}} ⊂ N are so that ∥x′_{i} − x_{i}∥_{2} ≤ ε then {x′_{i}}_{i} generates a non-injective von Neumann subalgebra in N.

Since N′_{0} ∩ N^{ω} is diffuse, it follows by induction that there exists a sequence of mutually commuting, τ-independent two dimensional abelian *-subalgebras A_{n} ⊂ N, with minimal projections of trace 1/2, such that if we denote by B_{n} = A_{1} ∨ A_{2} ∨ ...A_{n} then

∥E_{A′_{n+1}∩N}(x_{i}) − x_{i}∥_{2} < ε/2^{n+1}, ∀i.

But then we also have

∥E_{B′_{n}∩N}(x_{i}) − E_{B′_{n}∩N}(E_{A′_{n+1}∩N}(x_{i}))∥_{2} < ε/2^{n+1}, ∀1 ≤ i ≤ n,

Since E_{B′_{n}∩N} ∘ E_{A′_{n+1}∩N} = E_{B′_{n+1}∩N}, if we denote by A = ∨_{n}B_{n} and take into account that E_{A′∩N} = lim_{n} E_{B′_{n}∩N} (see e.g., [Po]), then by triangle inequalities we get

∥x_{i} − E_{A′∩N}(x_{i})∥_{2} ≤ ε, ∀i.

Thus, if we take N_{1} to be the von Neumann algebra generated by

x′_{i} = E_{A′∩N}(x_{i}), 1 ≤ i ≤ n,

then N_{1} satisfies the required conditions.

To get N_{1} to be a factor when N_{0} is a factor, we take at the same time with the elemnts x_{i} some additonal finite sets of unitaries, at each step of the induction argument, such that averaging of words of fixed length of elements previously chosen are close to scalars. Q.E.D.

Remarks 6.5. 1^{∘}. There is a striking analogy between the symmetry structure of the free group algebras and of the hyperfinite type II_{1} factor, with the free group factor L(𝔽_{∞}) seemingly having an edge....

Thus, all discrete groups act freely on the factor L(𝔽_{∞}), by free Bernoulli shifts. This should be compared with the fact that all groups act freely on the hyperfinite factor, by Bernoulli shifts. Moreover, the first examples of embeddings of arbitrary finite groups G in some AutN/IntN, with a given arbitrary scalar 3-cocycle, were constructed in ([Su]) on a “free-like” factor. Soon after, such examples were constructed in ([J1]) for arbitrary amenable groups G, on the hyperfinite type II_{1} factor N. In fact all “amenable symmetries” are now known to be present on the hyperfinite II_{1} factor by ([Po8, 9]).

All “finite symmetries” were shown to act on the free group factors L(𝔽_{n}), ∀2 ≤ n ≤ ∞, in ([Ra2]). In ([ShU]) it has been shown that there exist prime actions of any quantum group SU_{q}(n) on the factor L(𝔽_{∞}). More generally, it was recently proved in ([ShU]) that any symmetry “acts” on the free group factor L(𝔽_{∞}). In fact, by ([PoSh]) it seems that any kind of symmetry-phenomenon (the word being taken in a very broad sense) that can be realized on some type II_{1} factor, can be realized on L(𝔽_{∞}) as well. Theorem 6.1 above, which parallels the “hyperfinite” results 5.5-5.8, seems to confirm this fact. Moreover, it seems that L(𝔽_{∞}) is the unique factor having such a universality property with respect to symmetries.

2^{∘}. Related to the comments above, we have already mentioned in Remark 5.10.3^{∘} Choda’s result showing that there exist actions of SL(n, ℤ), n ≥ 3 on the hyperfinite factor such that the associated cross product algebra has the property T (in the sense of ([CJ]). Yet we do not know to construct an action of a property T group on L(𝔽_{∞}) that would give rise to a property T factor, via cross-product construction. Same for L(𝔽_{n}), 2 ≤ n < ∞.

In fact, while the free group algebras with finitely many generators apriori seem to have lesser symmetries, it seems more “feasable” to construct such rigid actions of property T groups on L(𝔽_{n}) than on L(𝔽_{∞}). On the other hand, there is a lack of examples of actions of arbitrary groups G on L(𝔽_{n}) that would behave as the free Bernoulli shifts on L(𝔽_{∞}) (for instance, actions of G on L(𝔽_{n}) that would implement a multiple of the left regular representation of G on ℓ^{2}(𝔽_{n}), as in [J2], while not normalizing any non-trivial hyperfinite subalgebra of L(𝔽_{n}), as in [Po3]).

3^{∘}. Related to the proof of Theorem 6.1 above and to (Lemma 4.3.2 in [Po4]), the following question on the group of automorphisms of the free group factors seems natural to ask:

Is Aut(L(𝔽_{n})), 2 ≤ n ≤ ∞, path-wise connected?

One should note that AutR does have this property, i.e., it is path-wise connected. Indeed, this follows trivially by the classification of single automorphisms in ([C2, 3]), which reduces the problem to some model automorphisms and ultimately to aperiodic such automorphisms, as follows:

Any automorphism θ ∈ AutR is conjugate to an inner perturbation of a model automorphism ρ_{0}. But Ad(u) ∘ ρ_{0} is path-wise connected to ρ_{0} and ρ_{0} ∼ ρ_{0} ⊗ id_{R}. Taking the “model” aperiodic automorphism to be an irrational rotation ρ_{1}, it follows that ρ_{1} is path-wise connected to id_{R}. Thus, ρ_{0} follows path-wise connected to ρ_{0} ⊗ ρ_{1}. But the latter is aperiodic, so it is connected to ρ_{1}, thus to id_{R}.

Appendix.

A.1. Definition of property T. We recall here a well known equivalent formulation of the property T for groups (see e.g., [DKi] or [dHV]; see also Lemma 4.1.5 in [Po4] for a general such statement).

A.1.1. Lemma. A group G has the property T of Kazhdan if and only if there exist g_{1}, g_{2}, ..., g_{n} ∈ G, ε_{0} > 0 and K > 0 such that for any representation π of G on a Hilbert space H and any vector ξ ∈ H satisfying

∥π(g_{i})ξ − ξ∥ ≤ ε_{0}∥ξ∥, ∀i,

one has

sup_{g∈G}∥π(g)ξ − ξ∥ ≤ Kmax_{1≤i≤n}∥π(g_{i})ξ − ξ∥

A.2. Group algebra extensions. An important class of cocycle actions can be obtained from inclusions of group von Neumann algebras, as follows. Let 1 → H ↪ K → G → 1 be an exact sequence of discrete groups and note that one has:

A.2.1. Lemma. Let H ⊂ K be an inclusion of discrete groups. Then L(H)′ ∩ L(K) = ℂ if and only if K has infinite conjugacy classes relative to H, i.e., {hgh^{−1} | h ∈ H} is an infinite set ∀g ∈ G, g ≠ e.

Proof. The proof is identical to the classical case for single groups ([MvN], [Di]). Q.E.D.

For each g ∈ G let k(g) ∈ K be a lifting of g in K and denote by σ_{g} ∈ Aut L(H) the automorphism σ_{g}(x) = u_{k(g)}xu_{k(g)}^{∗}, x ∈ L(H). Also for each g_{1}, g_{2} ∈ G denote by v_{g_{1},g_{2}} the unitary element u_{h} ∈ L(H) ⊂ L(K), where h = k(g_{1})k(g_{2})k(g_{1}g_{2})^{−1} ∈ H. Then (σ, v) is clearly a properly outer cocycle action and L(H) ⊂ L(K) can be viewed as the cocycle cross product N ⊂ N ⋊_{σ,v} G, where N = L(H) (see [NT]).

Along these lines, a particularly interesting example is the Connes-Jones cocycle (cf. [CJ]): Let G be an arbitrary countable discrete group with generators g_{1}, g_{2}, …, g_{n}, where 1 ≤ n ≤ ∞, but with G ≠ 𝔽_{n}. Let 𝔽_{n} → G → 1 be the corresponding presentation of (G; g_{1}, …, g_{n}). Let H = ker(𝔽_{n} → G) and note that H ≃ 𝔽_{nm}, where m = |G|. Then 𝔽_{n} has infinite conjugacy class relative to its normal subgroup H. Thus (L(H) ⊂ L(𝔽_{n})) = (L(H) ⊂ L(H) ⋊_{σ,v} G) for an appropriate properly outer cocycle action (σ, v) of G on L(H), with H ≃ 𝔽_{∞} whenever G is an infinite group with generators g_{1}, …, g_{n} not all mutually free.

By ([CJ]), the Connes-Jones cocycle is non-vanishing when G is an infinite group with the property T. The proof is based on the observation that L(G) cannot be embedded in L(𝔽_{n}), due to Haagerup’s compact approximation property for L(𝔽_{n})

([H]). The same argument actually works for the group G = SL(2, ℤ) ⋊ ℤ^{2}, which doesn’t have the property T. For the proof, one uses the rigidity of the inclusion ℤ^{2} ⊂ G to conclude that if one takes the approximation of the identity on L(𝔽_{n}) by compact, unital, trace preserving maps then the identity on L(ℤ^{2}) follows close to a compact operator, uniformly on all the unitaries in L(ℤ^{2}), a contradiction (see [Po4] for a complete argument along these lines).

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E-mail address: popa@math.ucla.edu