On the structure of Banach algebras associated with automorphisms

We study the structure of the Banach algebra $B(A, T_g )$ generated by a Banach algebra $A$ of operators in a Banach space $D$ and a group $\{ T_g \}_{g \in G}$ of isometries of $D$ such that $T_g A T^{-1}_g = A$. We analyze relations between the existence of an expectation from $B(A, T_g )$ to $A$, topological (metric) freeness of the action by automorphisms of the algebra $A$ induced by $T_g$, and the dual action of the group $G$ on $B(A, T_g )$. The obtained results are applied to the description of the structure of Banach algebras generated by ``weighted composition operators'' in various spaces.