$C^*$-algebras generated by mappings. Criterion of irreducibility

We study the operator algebra associated with a self-mapping $\varphi $ on a countable set $ X $ which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding $ l^ 2(X) $. We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic $C^*$-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.