$K$-Groups of some subalgebras of the Toeplitz algebra

In this paper it is discussed $K$-groups of $C^*$-subalgebras of the Toeplitz algebra generated by inverse subsemigroups of the bicyclic semigroup. For these algebras constructed inductive limit of inductive sequence of $K$-groups, which are generated by the corresponding inductive sequence of $C^*$-algebras.