  In the present paper we study the structure of a Banach algebra B(A, T_g)
generated by a certain Banach algebra $A$ of operators acting 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 investigate the interrelations between the existence of the expectation of
B(A, T_g) onto A, topological freedom of the automorphisms of A induced by T_g
and the dual action of the group G on B(A, T_g). The results obtained are
applied to the description of the structure of Banach algebras generated by
'weighted composition operators' acting in various spaces.
