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

TL;DR

This paper investigates the structure of rigid subalgebras within von Neumann algebras under deformation, establishing maximality and intersection properties that deepen understanding of rigidity phenomena in operator algebras.

Contribution

It proves that rigid subalgebras under mixing s-malleable deformation are contained in uniquely maximal rigid subalgebras, and explores implications for algebra generation and group properties.

Findings

Rigid subalgebras are contained in uniquely maximal rigid subalgebras.

The algebra generated by intersecting rigid subalgebras is itself rigid.

Applications to group von Neumann algebras with positive first $L^2$-Betti number.

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 $\widetilde M \supseteq M$ by an action $\alpha: \mathbb{R} \to {\rm Aut}(\widetilde M)$, while simultaneously containing subalgebras $Q$ {\it rigid} with respect to that deformation, that is, such that $\alpha_t \to {\rm id}$ 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 $\beta^{1}_{(2)}(G) > 0$, then $L(G)$ cannot be generated by two property $(T)$ subalgebras with diffuse intersection; however, the result is most striking when the family is infinite.