Classification of crossed-product $C^ *$-algebras associated with characters on free groups

Abstract:

We study the classification problem of crossed-product ${C^ * }$-algebras of the form $C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf {Z}}$, where $G$ is a discrete group, $\chi$ is a one-dimensional character of $G$, and ${\alpha _\chi }$ is the unique $*$-automorphism of $C_r^ * (G)$ such that if $U$ is the left regular representation of $G$, then ${\alpha _{\chi }(U_{g})=\chi (g)U_{g}}$, $g \in G$. When ${G = F_{n}}$, the free group on $n$ generators, we have a complete classification of these crossed products up to $*$-isomorphism which generalizes the classification of irrational and rational rotation ${C^ * }$-algebras. We show that these crossed products are determined by two $K$-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural $\operatorname {Aut} ({F_n})$-action, and that these two orbit invariants completely classify the characters up to automorphisms of ${F_n}$. The classification of crossed products follows from these results. We also consider the same problem for $G$ some other groups.