Random $C^{\ast }$-subalgebras

Given a discrete countable group $\mathrm{\Gamma }$, we introduce the concept of random subalgebras of the reduced group $C^{\ast }$-algebra $C_r^{\ast }(\mathrm{\Gamma })$. This extends the existing theory of invariant random subgroups and the one of invariant random subalgebras of the group von Neumann algebra $\mathrm{L}(\mathrm{\Gamma })$. As applications, we provide a characterisation of $C^{\ast }$-simplicity, a condition for amenability of the group and show the existence and uniqueness of a maximal nuclear $\mathrm{\Gamma }$-invariant unital $C^{\ast }$-subalgebra of $C_r^{\ast }(\mathrm{\Gamma })$. Moreover, we give a topological characterisation of exactness of any separable unital $C^{\ast }$-algebra. Finally, we study exotic random subalgebras of $C_r^{\ast }(\mathrm{\Gamma })$ and super-exotic random subalgebras of $\mathrm{L}(\mathrm{\Gamma })$, that is, random subalgebras which do not arise purely from random subgroups of $\mathrm{\Gamma }$ and not from random subalgebras of $C_r^{\ast }(\mathrm{\Gamma })$, respectively.