Some permanence properties of C∗-unique groups

We will study some permanence properties of C∗-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C∗-unique group. Moreover, any amenable discrete group is a retract of a discrete C∗-unique group.