Classification of semicrossed products of finite-dimensional $C^ \ast$-algebras

Abstract:

Let $\mathfrak {A}$, $\mathfrak {B}$ be finite-dimensional ${C^*}$-algebras with automorphisms $\alpha$, $\beta$, respectively. Then the semicrossed products ${{\mathbf {Z}}^ + }{ \times _\alpha }\mathfrak {A}$, ${{\mathbf {Z}}^ + }{ \times _\beta }\mathfrak {B}$ are isomorphic iff there is an isomorphism $\psi :\mathfrak {A} \to \mathfrak {B}$ and a unitary $U \in \mathfrak {B}$ such that $\beta \circ \psi = (\operatorname {Ad} U)\psi \circ \alpha$.