Group C*-algebras satisfying Kadison's conjecture
Volume 96 / 2011
Abstract
We tackle R. V. Kadison's similarity problem (i.e. any bounded representation of any unital C$^*$-algebra is similar to a $^*$-representation), paying attention to the class of C$^*$-unitarisable groups (those groups $G$ for which the full group C$^*$-algebra C${^*}(G)$ satisfies Kadison's problem) and thereby we establish some stability results for Kadison's problem. Namely, we prove that $A\otimes_{\rm min} B$ inherits the similarity problem from those of the C$^*$-algebras $A$ and $B$, provided $B$ is also nuclear. Then we prove that $G/\Gamma$ is C$^*$-unitarisable provided $G$ is C$^*$-unitarisable and $\Gamma$ is a normal subgroup; and moreover, if $G/\Gamma$ is amenable and $\Gamma$ is C$^*$-unitarisable, so is the whole group $G$ ($\Gamma$ a normal subgroup).
