Isometric representations of totally ordered semigroups

Abstract

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 × ℤ and prove that if S induces total order on G, then there exist at least two unitarily not equivalent irreducible isometrical representations of S. And if the order is lexicographical-product order, then all such representations are unitarily equivalent.