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

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