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

In this paper we consider a $ C^*$-subalgebra of the algebra of all bounded operators $B(l^2(X))$ on the Hilbert space $l^2(X)$ with one generating element $T_\varphi$ induced by a mapping $\varphi\colon X\to X$ of the 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.