One class of $C^*$-algebras generated by a family of partial isometries and multiplicators

We consider a $C^*$-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. This algebra is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The operators of partial isometries satisfy relations defined by a prescribed map on the set. We show that the considered algebra is $\mathbb Z$-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.