AF-subalgebras of a C*-algebra generated by a mapping

In this paper we consider a C*-subalgebra of the algebra of all bounded operators B(l²(X)) on the Hilbert space l²(X)) with one generating element T _φ induced by a mapping φ of a set X into itself. We prove that such a C* -algebra has an AF-subalgebra and establish commutativity conditions for the latter. We prove that a C* -algebra generated by a mapping produces a dynamic system such that the corresponding group of automorphisms is invariant on elements of the AF- subalgebra.