$C^*$-Algebras Generated by Mappings

In the paper, some properties of a singly generated $C^*$-subalgebra of the algebra of all bounded operators $B(l^2(X))$ on the Hilbert space $l^2(X)$ with the generator $T_\varphi$ induced by a mapping $\varphi$ of an infinite set $X$ into itself are investigated. A condition on $\varphi$ is presented under which the operator $T_\varphi$ is continuous, and it is proved that, if this is the case, then the study of these algebras can be reduced to that of $C^*$-algebras generated by a finite family of partial isometries of a special form. A complete description of the $C^*$-algebras generated by an injective mapping on $X$ is given. Examples of $C^*$-algebras generated by noninjective mappings are treated.