Abstract
The algebra under study belongs to the class of operator algebras generated by a family of partial isometries, satisfying some relations on the initial and final projections. In turn, this family is uniquely determined by a self-mapping of a countable set. In the paper we consider a situation when isometry family generates an inverse semigroup. It is shown that in this (and only in this) case the corresponding C*-algebra has a nontrivial commutative AF-subalgebra, generated by a semi-lattice of projections of inverse semigroup. All invariant subspaces of the mentioned C*-algebra and its irreducible representations are described.
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