Thomas Sinclair

ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS

Abstract This paper contains a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance, we show that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid. There is also the following product version of this result: any maximal abelian ⋆ -subalgebra of any II 1 factor associated with a finite product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with a cocycle superrigidity result of Ioana, it follows that compact actions by finite products of lattices in S p ( n , 1 ) , n ≥ 2 , are virtually W ∗ -superrigid.

On cocycle superrigidity for Gaussian actions

We present a general setting to investigate U_fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some U_fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be U_fin-cocycle superrigid.

Strong solidity of group factors from lattices in SO(n,1) and SU(n,1)

Abstract We show that the group factors LΓ, where Γ is an ICC lattice in either SO ( n , 1 ) or SU ( n , 1 ) , n ⩾ 2 , are strongly solid in the sense of Ozawa and Popa (2010) [13] . This strengthens a result of Ozawa and Popa (2010) [14] showing that these factors do not have Cartan subalgebras.

Inner amenability for groups and central sequences in factors

We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin [Groups with hyperbolically embedded subgroups. Algebr. Geom. Topol. 13 (2013), 2635–2665], our result recovers that mapping class groups and

Robinson forcing and the quasidiagonality problem

\text{Out}(\mathbb{F}_{n})

On the structural theory of

are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann [On rings of operators IV. Ann. of Math. (2) 44 (1943), 716–808].

{\rm II}_1

We introduce weakenings of two of the more prominent open problems in the classification of

factors of negatively curved groups

\mathrm{C}^*

On Kirchberg's embedding problem

-algebras, namely the quasidiagonality problem and the UCT problem. We show that the a positive solution of the conjunction of the two weaker problems implies a positive solution of the original quasidiagonality problem as well as allows us to give a local, finitary criteria for the MF problem, which asks whether every stably finite

The theory of tracial von Neumann algebras does not have a model companion

\mathrm{C}^*