Isometric Representations of Totally Ordered Semigroups

TL;DR

This paper characterizes when semigroups induce isomorphic C*-algebras via isometric representations, linking algebraic properties of the semigroup with the structure of its representations and the order on the group.

Contribution

It proves the reverse of a known theorem, establishing that isomorphic C*-algebras imply the semigroup is a positive cone, and analyzes representations for specific ordered groups.

Findings

C*-algebras are isomorphic iff S is a positive cone of G

Multiple inequivalent irreducible representations exist for certain orders on Z×Z

All representations are equivalent under lexicographical order

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\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.