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April 10, 2020

Maximal Rigid Subalgebras of Deformations

We show that 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 positive first L$^2$-Betti number cannot be generated by two property (T) subalgebras with diffuse intersection. This is joint with Dan Hoff, Ben Hayes, and Rolando de Santiago.