On the structure of $C^*$-algebras generated by representations of the elementary inverse semigroup

The class of $C^*$-algebras generated by the elementary inverse semigroup and being deformations of the Toeplitz algebra has been introduced and studied. The properties of these algebras have been investigated. All their irreducible representations and automorphism groups have been described. These algebras have been proved to be $\mathbb Z$-graded $C^*$-algebras. For a certain class of algebras in the family under consideration the compact quantum semigroup structure has been constructed.