Stability results for $C^*$-unitarizable groups

Abstract

‎We say that a locally compact group $G$ is $C^*$-unitarizable if its‎ ‎full group $C^*$-algebra $C^*(G)$ satisfies Kadison's similarity‎ ‎problem (SP)‎, ‎i.e. every bounded representation of $C^*(G)$ on a‎ ‎Hilbert space is similar to a *-representation‎. ‎We prove that‎ ‎locally compact and unitarizable groups are $C^*$-unitarizable‎. ‎For‎ ‎discrete groups‎, ‎we prove that $C^*$-unitarizable passes to‎ ‎quotients‎. ‎Moreover‎, ‎a given discrete group is $C^*$-unitarizable‎ ‎whenever we can find a normal and $C^*$-unitarizable subgroup with‎ ‎amenable quotient‎.