Invariant C⁎-subalgebras of the reduced group C⁎-algebra

Let $\Gamma$ be a countable discrete group. We say that $\Gamma$ has $C^*$-invariant subalgebra rigidity (ISR) property if every $\Gamma$-invariant $C^*$-subalgebra $\mathcal{A}\le C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for some normal subgroup $N\triangleleft\Gamma$. 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$^*$-ISR property only if $\Gamma$ is simple amenable or $C^*$-simple.