Type of some nuclear subalgebras of the Toeplitz algebra generated by the inverse subsemigroups of a bicyclic semigroup

Abstract

We construct a $Z$-graded structure of the Toeplitz algebra and consider nuclear $C\ast$ -subalgebras of the Toeplitz algebra generated by the inverse subsemigroups of the bicyclic semigroup. The types of these algebras with respect to the Toeplitz algebra are determined. In addition, it is shown, that considered algebras are equipped with the structrure of Hilbert $C\ast$ -modules.