arXiv subjects
Publications and source records attributed to V. H. Tepoyan.
We study inverse $π$-extensions of the semigroup $\mathbb{Z}_+$. It is shown that $π$-extension of the semigroup $\mathbb{Z}_+$ is inverse, iff its $π$-extension coincides with $π(\mathbb{Z}_+)$. The existence of a non-inverse $π$-extension for semigroup $\mathbb{Z}_+$ is proved.
We introduce the notion of π-extension of the semigroup \mathbb{Z}_+ and study the extensions of the Toeplitz algebras by isometric operators. We show that when the action of the Toeplitz algebra is irreducible all such extensions generate the same algebra, i.e. there are no non-trivial extensions of the Toeplitz algebra. Also we provide the examples of the non-trivial extensions of the Toeplitz algebra in case its representation is reducible.
Let S be a subsemigroup of an abelian torsion-free group G. If S is a positive cone of G, then all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic. Proved by Murphy, this statement generalized the well-known theorems of Coburn and Douglas. In this note we prove the reverse. If all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic, then S is a positive cone of G. Also we consider G = Z\times Z and prove that if S induces total order on G, then there exist at least two unitarily not equivalent irreducible isometrical representation of S. And if the order is lexicographical-product order, then all such representations are unitarily equivalent.