On the structure of $C^*$-algebra generated by a family of partial isometries and multipliers

In the paper we consider an operator algebra generated by a family of partial isometries associated with a self-mapping on a countable set and by multipliers. An action of the unit circle on this algebra is specified that determines its $\mathbb{Z}$-grading. Under some conditions on the mapping the algebra is isomorphic to the crossed product of its fixed point subalgebra and the semigroup $\mathbb{N}$