Conditional expectations in $C^{\ast }$-crossed products

Abstract:

Let $(A, G, \alpha )$ be a ${C^{\ast }}$-dynamical system. Let $B$ be a ${C^{\ast }}$-subalgebra of $A$ and $P$ be a conditional expectation of $A$ onto $B$ such that ${\alpha _t}P = P{\alpha _t}$ for each $t \in G$. Then it is proved that there exists a conditional expectation of ${C^{\ast }}(G, A, \alpha )$ onto ${C^{\ast }}(G, B, \alpha )$. In particular, if $G$ is amenable and $A$ is unital, then there always exists a conditional expectation of ${C^{\ast }}(G, A, \alpha )$ onto ${C^{\ast }}(G)$. Some related results are also obtained.