A II1_1 factor approach to the Kadison-Singer problem

We show that the Kadison-Singer problem, asking whether the pure states of the diagonal subalgebra \ell^\infty\Bbb N\subset \Cal B(\ell^2\Bbb N) have unique state extensions to \Cal B(\ell^2\Bbb N), is equivalent to a similar statement in II$_1$ factor framework, concerning the ultrapower inclusion $D^\omega \subset R^\omega$, where $D$ is the Cartan subalgebra of the hyperfinite II$_1$ factor $R$, and $\omega$ is a free ultraflter. While we do not settle the problem in this latter form, we prove that if $A$ is any singular maximal abelian subalgebra of $R$, then the inclusion $A^\omega \subset R^\omega$ does satisfy the Kadison-Singer property.Comment: Final version, to appear in Comm Math Phys: "Added in the Proof" at end of the Introd., on the recent solution to the classic Kadison-Singer by Marcus-Spielman-Strivastava (arXiv:1306.3969); one more characterization of singular MASAs in Thm. 0.2 (resp. Thm. 5.2); more details + complements in proof of 4.1; two sub-sections in Sec. 5 removed (to appear elsewhere

On the classification of inductive limits of II1_1 factors with spectral gap

We consider II$_1$ factors $M$ which can be realized as inductive limits of subfactors, $N_n \nearrow M$, having spectral gap in $M$ and satisfying the bi-commutant condition $(N_n'\cap M)'\cap M=N_n$. Examples are the enveloping algebras associated to non-Gamma subfactors of finite depth, as well as certain crossed products of McDuff factors by amenable groups. We use deformation/rigidity techniques to obtain classification results for such factors.Comment: 14 page

On Ozawa's Property for Free Group Factors

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.Comment: Final form; paper appeared in Int. Math. Res. Notices, 200

Asymptotic orthogonalization of subalgebras in II1_1 factors

Let $M$ be a II$_1$ factor with a von Neumann subalgebra $Q\subset M$ that has infinite index under any projection in $Q'\cap M$ (e.g., $Q$ abelian; or $Q$ an irreducible subfactor with infinite Jones index). We prove that given any separable subalgebra $B$ of the ultrapower II$_1$ factor $M^\omega$, for a non-principal ultrafilter $\omega$ on $\Bbb N$, there exists a unitary element $u\in M^\omega$ such that $uBu^*$ is orthogonal to $Q^\omega$.Comment: Final version. To appear in Publ RIM

Cocycle and Orbit Equivalence Superrigidity for Malleable Actions of w-Rigid Groups

We prove that if a countable discrete group $\Gamma$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $\Gamma= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $\Gamma = H \times H'$ 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 $\Gamma \curvearrowright X$ of $\Gamma$ on a probability space $(X,\mu)$ which is weak mixing on $H$ and {\it s-malleable} (e.g. the Bernoulli action $\Gamma \curvearrowright [0,1]^\Gamma$) is cohomologous to a group morphism of $\Gamma$ into \Cal V. We use the case \Cal V discrete of this result to prove that if in addition $\Gamma$ has no non-trivial finite normal subgroups then any orbit equivalence between $\Gamma \curvearrowright X$ and a free ergodic measure preserving action of a countable group $\Lambda$ is implemented by a conjugacy of the actions, with respect to some group isomorphism $\Gamma \simeq \Lambda$.Comment: Final version; paper appeared in Invent Math Vol 170 (2007), 243-29

Independence properties in subalgebras of ultraproduct II1_1 factors

Let $M_n$ be a sequence of finite factors with $\dim(M_n)\rightarrow \infty$ and denote $\text{\bf M}=\Pi_\omega M_n$ their ultraproduct over a free ultrafilter $\omega$. We prove that if $\text{\bf Q}\subset \text{\bf M}$ is either an ultraproduct $\text{\bf Q}=\Pi_\omega Q_n$ of subalgebras $Q_n\subset M_n$, with $Q_n \not\prec_{M_n} Q_n'\cap M_n$, $\forall n$, or the centralizer $\text{\bf Q}=B'\cap \text{\bf M}$ of a separable amenable *-subalgebra $B\subset \text{\bf M}$, then for any separable subspace $X\subset \text{\bf M}\ominus (\text{\bf Q}'\cap \text{\bf M})$, there exists a diffuse abelian von Neumann subalgebra in $\text{\bf Q}$ which is {\it free independent} to $X$, relative to $\text{\bf Q}'\cap \text{\bf M}$. Some related independence properties for subalgebras in ultraproduct II$_1$ factors are also discussed.Comment: Some minor corrections and additions; final version (33 pages), with corrections and 2.2.3 added; to appear in JF

On the Fundamental Group of Type II1_1 Factors

We present here a shorter version of the proof of a result from our paper ``On a class of type II$_1$ factors with Betti numbers invariants'', showing that the von Neumann factor associated with the group $\Bbb Z^2 \rtimes SL(2, \Bbb Z)$ has trivial fundamental group

On spectral gap rigidity and Connes invariant χ(M)\chi(M)

We calculate Connes' invariant $\chi(M)$ for certain II$_1$ factors $M$ that can be obtained as inductive limits of subfactors with spectral gap, then use this to answer a question he posed in 1975, on the structure of McDuff factors $M$ with $\chi(M)=1$.Comment: 8 page

A Unique Decomposition Result for HT Factors with Torsion Free Core

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 $L_\alpha(\Bbb Z^2) \rtimes \Gamma$, for torsion free, non-amenable subgroups $\Gamma\subset SL(2, \Bbb Z)$ and $\alpha = e^{2\pi i t}$, $t \not\in \Bbb Q$, implies isomorphism of the corresponding groups $\Gamma$.Comment: 8 pages; minor corrections 02/17/04, 03/17/0

Some computations of 1-cohomology groups and construction of non orbit equivalent actions

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 $\Lambda$ we construct free ergodic measure preserving actions $\sigma_\Lambda$ of $G$ on the probability space such that the 1'st cohomology group of $\sigma_\Lambda$, $H^1(\sigma_\Lambda)$, is equal to Char$(G) \times \Lambda$. 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.Comment: 24 pages (final version, with slight additions and change of title on 09/20/04