#### Vol. 14, No. 7, 2021

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Maximal rigid subalgebras of deformations and $L^{2}$-cohomology

### Rolando de Santiago, Ben Hayes, Daniel J. Hoff and Thomas Sinclair

Vol. 14 (2021), No. 7, 2269–2306
##### Abstract

In the past two decades, Sorin Popa’s breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $\stackrel{˜}{M}\supseteq M$ by an action $\alpha :ℝ\to Aut\left(\stackrel{˜}{M}\right)$, while simultaneously containing subalgebras $Q$ rigid with respect to that deformation, that is, such that ${\alpha }_{t}\to id$ uniformly on the unit ball of $Q$ as $t\to 0$. However, it has remained unclear how to exploit the interplay between distinct rigid subalgebras not in specified relative position.

We show that in fact, any diffuse subalgebra which is rigid with respect to a mixing s-malleable deformation is contained in a subalgebra which is uniquely maximal with respect to being rigid. In particular, the algebra generated by any family of rigid subalgebras that intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members was the motivation for this work, showing for example that if $G$ is a countable group with ${\beta }_{\left(2\right)}^{1}\left(G\right)>0$, then $L\left(G\right)$ cannot be generated by two property-(T) subalgebras with diffuse intersection; however, the result is most striking when the family is infinite.

##### Keywords
deformation, rigidity, property (T), quasinormalizer
##### Mathematical Subject Classification 2010
Primary: 46L36, 46L10
Secondary: 37A55
##### Milestones
Received: 2 November 2019
Revised: 2 May 2020
Accepted: 15 June 2020
Published: 10 November 2021
##### Authors
 Rolando de Santiago Department of Mathematics Purdue University West Lafayette, IN United States Ben Hayes Department of Mathematics University of Virginia Charlottesville, VA United States Daniel J. Hoff Department of Mathematics University of California Los Angeles Los Angeles, CA United States Thomas Sinclair Department of Mathematics Purdue University West Lafayette, IN United States