Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra

Abstract

Let $\Gamma$ be a countable discrete group. We say that $\Gamma$ has $C^{\ast }$-invariant subalgebra rigidity (ISR) property if every $\Gamma$-invariant $C^{\ast }$-subalgebra $\mathcal{A}\le C_r^{\ast }(\Gamma )$ is of the form $C_r^{\ast }(N)$ for some normal subgroup $N◃\Gamma$. We show that all torsion-free, non-amenable (cylindrically) hyperbolic groups with property-AP and a finite direct product of such groups have this property. We also prove that an infinite group $\Gamma$ has the C${}^{\ast }$-ISR property only if $\Gamma$ is simple amenable or $C^{\ast }$-simple.

Final version: all the referee's suggestions have been implemented

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• Charmenability of higher rank arithmetic groups
• Subalgebras, subgroups, and singularity
• On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras
• On the amenable subalgebras of group von Neumann algebras