$C^*$-algebras generated by mappings. Classification of invariant subspaces

We continue the study of 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 in a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projectors. Earlier we formulated the irreducibility criterion of such algebras. With its help we will examine the structure of the the corresponding Hilbert space. We will show that for a reducible algebra the underlying Hilbert space is represented either as an infinite sum of invariant subspaces or in the form of a tensor product of finite-dimensional Hilbert space and $ l ^ 2 (\mathbb{Z})$. In the first case we give the conditions when the studied algebra has an irreducible representation into a $ C^*$-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.