Almost commuting matrices and stability for product groups

Title: Almost commuting matrices and stability for product groups

Speaker: Adrian Ioana (UCSD)

Abstract: I will present a result showing that the direct product group $G=\mathbb F_2\times\mathbb F_2$ is not Hilbert-Schmidt stable. This means that G admits a sequence of asymptotic homomorphisms (with respect to the normalized Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms.  While this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. I will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices A,B such that A almost commutes with B and B* (in the normalized Hilbert-Schmidt norm), but there are no matrices A’,B’ close to A,B such that A’ commutes with B’ and B’*.

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