arXiv subjects
Publications and source records attributed to Sorin Popa.
This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors.
We show that given any subgroup F of R_+ which is either countable or belongs to a certain "large" class of uncountable subgroups, there exist continuously many free ergodic probability measure preserving actions σ_i of the free group with infinitely many generators such that their associated group measure space II_1 factors M_i and orbit equivalence relations R_i have fundamental group equal to F and with M_i (respectively R_i) stably non-isomorphic. Moreover, these actions can be taken so that R_i has no outer automorphisms and any automorphism of M_i is unitary conjugate to an automorphism that acts trivially on $L^\infty(X_i) \subset M_i$.
We prove that if a countable discrete group $Γ$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $Γ= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $Γ= H \times H'$ with $H$ an infinite Kazhdan group and $H'$ arbitrary), and $\Cal V$ is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. $\Cal V$ countable discrete, or separable compact), then any $\Cal V$-valued measurable cocycle for a measure preserving action $Γ\curvearrowright X$ of $Γ$ on a probability space $(X,μ)$ which is weak mixing on $H$ and {\it s-malleable} (e.g. the Bernoulli action $Γ\curvearrowright [0,1]^Γ$) is cohomologous to a group morphism of $Γ$ into $\Cal V$. We use the case $\Cal V$ discrete of this result to prove that if in addition $Γ$ has no non-trivial finite normal subgroups then any orbit equivalence between $Γ\curvearrowright X$ and a free ergodic measure preserving action of a countable group $Λ$ is implemented by a conjugacy of the actions, with respect to some group isomorphism $Γ\simeq Λ$.
We prove that if a countable group $Γ$ contains infinite commuting subgroups $H, H'\subset Γ$ with $H$ non-amenable and $H'$ ``weakly normal'' in $Γ$, then any measure preserving $Γ$-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli $Γ$-action) is cocycle superrigid. If in addition $H'$ can be taken non-virtually abelian and $Γ\curvearrowright X$ is an arbitrary free ergodic action while $Λ\curvearrowright Y=\Bbb T^Λ$ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II$_1$ factors $L^\infty X \rtimes Γ\simeq L^\infty Y \rtimes Λ$ comes from a conjugacy of the actions.
We give a new proof of a result of Ozawa showing that if a von Neumann subalgebra $Q$ of a free group factor $L\Bbb F_n, 2\leq n\leq \infty$ has relative commutant diffuse (i.e. without atoms), then $Q$ is amenable.
We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,μ)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.
We consider $\text{\rm II}_1$ factors $L_μ(G)$ arising from 2-cocyles $μ\in \text{\rm H}^2(G,\Bbb T)$ on groups $G$ containing infinite normal subgroups $H \subset G$ with the relative property $\text{\rm(T)}$ (i.e. $G$ {\it w-rigid}). We prove that given any separable $\text{\rm II}_1$ factor $M$, the set of 2-cocycles $μ_{|H}\in \text{\rm H}^2(H,\Bbb T)$ with the property that $L_μ(G)$ is embeddable into $M$ is at most countable. We use this result, the relative property (T) of $\Bbb Z^2 \subset \Bbb Z^2 \rtimes Γ$ for $Γ\subset SL(2,\Bbb Z)$ non-amenable and the fact that every cocycle $μ_α\in {\text{\rm H}}^2(\Bbb Z^2,\Bbb T)\simeq \Bbb T$ extends to a cocycle on $\Bbb Z^2 \rtimes SL(2,\Bbb Z)$, to show that the one parameter family of II$_1$ factors $M_α(Γ)=L_{μ_α}(\Bbb Z^2 \rtimes Γ)$, $α\in \Bbb T$, are mutually non-isomorphic, modulo countable sets, and cannot all be embedded into the same separable II$_1$ factor. Other examples and applications are discussed.
We study equivalence relations and II_1 factors associated with (quotients of) generalized Bernoulli actions of Kazhdan groups. Specific families of these actions are entirely classified up to isomorphism of II_1 factors. This yields explicit computations of outer automorphism and fundamental groups. In particular, every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic II_1 factors.
We construct numerous continuous families of irreducible subfactors of the hyperfinite II$_1$ factor, which are non-isomorphic, but have all the same standard invariant. In particular, we obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II$_1$ factor, which have Jones index 6 and have all the same standard invariant with property (T). We exploit the fact that property (T) groups have uncountably many non-cocycle conjugate cocycle actions on the hyperfinite II$_1$ factor.
We consider two von Neumann subalgebras $\cl B_0$ and $\cl B$ of a type ${\rm{II}}_1$ factor $\cl N$. For a map $ϕ$ on $\cl N$, we define \[\|ϕ\|_{\infty,2}=\sup\{\|ϕ(x)\|_2\colon \|x\| \leq 1\},\] and we measure the distance between $\cl B_0$ and $\cl B$ by the quantity $\|{\bb E}_{\cl B_0}-{\bb E}_{\cl B}\|_{\infty,2}$. Under the hypothesis that the relative commutant in $\cl N$ of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the $\|\cdot \|_2$--norm. This hypothesis is satisfied, in particular, by masas and subfactors of trivial relative commutant. A general version with a slightly weaker conclusion is also proved. As a consequence, we show that if $\cl A$ is a masa and $u\in\cl N$ is a unitary such that $\cl A$ and $u\cl Au^*$ are close, then $u$ must be close to a unitary which normalizes $\cl A$. These qualitative statements are given quantitative formulations in the paper.
We consider cross-product II$_1$ factors $M = N\rtimes_σ G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $σ: G \to {\text{\rm Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=\Bbb Z^2 \rtimes SL(2, \Bbb Z)$. We deduce that for any countable group $S \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.
We prove that any isomorphism $θ:M_0\simeq M$ of group measure space II$_1$ factors, $M_0=L^\infty(X_0, μ_0) \rtimes_{σ_0} G_0$, $M=L^\infty(X, μ) \rtimes_σ G$, with $G_0$ containing infinite normal subgroups with the relative property (T) of Kazhdan-Margulis (i.e. $G_0$ {\it w-rigid}) and $G$ an ICC group acting by Bernoulli shifts $σ$, essentially comes from an isomorphism of probability spaces which conjugates the actions. Moreover, any isomorphism $θ$ of $M_0$ onto a ``corner'' $pMp$ of $M$, for $p\in M$ an idempotent, forces $p=1$. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and all isomorphisms between such factors come from isomorphisms of the corresponding groups. This settles a ``group measure space version'' of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations.
For each group $G$ having an infinite normal subgroup with the relative property (T) (for instance $G = H \times K$ where $H$ is infinite with property (T) and $K$ is arbitrary), and any countable abelian group $Λ$ we construct free ergodic measure preserving actions $σ_Λ$ of $G$ on the probability space such that the 1'st cohomology group of $σ_Λ$, $H^1(σ_Λ)$, is equal to Char$(G) \times Λ$. We deduce that $G$ has uncountably many non stably orbit equivalent actions. We also calculate 1-cohomology groups and show existence of ``many'' non stably orbit equivalent actions for free products of groups as above.
For each $2 \leq n \leq \infty$, we construct an uncountable family of free ergodic measure preserving actions $α_t$ of the free group $\Bbb F_n$ on the standard probability space $(X, μ)$ such that any two are non orbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid'' (in the sense of [Po01]), with the II$_1$ factors $L^\infty(X, μ)\rtimes_{α_t} \Bbb F_n$ mutually non stably isomorphic (even non-stably isomorphic) and in the class $\Cal H\Cal T_{_{s}}.$
We prove that II$_1$ factors $M$ have a unique (up to unitary conjugacy) cross-product type decomposition around ``core subfactors'' $N \subset M$ satisfying the property HT and a certain ``torsion freeness'' condition. In particular, this shows that isomorphism of factors of the form $L_α(\Bbb Z^2) \rtimes Γ$, for torsion free, non-amenable subgroups $Γ\subset SL(2, \Bbb Z)$ and $α= e^{2πi t}$, $t \not\in \Bbb Q$, implies isomorphism of the corresponding groups $Γ$.
We prove that the notion of relative property (T) (or rigidity) for inclusions of finite von Neumann algebras defined in [Po1] is equivalent to a weaker property, in which no ``continuity constants'' are required. The proof is by contradiction and uses infinite products of completely positive maps, regarded as correspondences.
We prove that a type II$_1$ factor $M$ can have at most one Cartan subalgebra $A$ satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class $\Cal H \Cal T$ of factors $M$ having such Cartan subalgebras $A \subset M$, the Betti numbers of the standard equivalence relation associated with $A \subset M$ ([G2]), are in fact isomorphism invariants for the factors $M$, $β^{^{HT}}_n(M), n\geq 0$. The class $\Cal H\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $β^{^{HT}}_n(M^t)= β^{^{HT}}_n(M)/t, \forall t>0$, and a K{ü}nneth type formula. An example of a factor in the class $\Cal H\Cal T$ is given by the group von Neumann factor $M=L(\Bbb Z^2 \rtimes SL(2, \Bbb Z))$, for which $β^{^{HT}}_1(M) = β_1(SL(2, \Bbb Z)) = 1/12$. Thus, $M^t \not\simeq M, \forall t \neq 1$, showing that the fundamental group of $M$ is trivial. This solves a long standing problem of R.V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.
We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to Π} (X_0, μ_0)_g$ for $(X_{0},μ_{0})$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.