Isometric Representations of Totally Ordered Semigroups

Details Isometric Representations of Totally Ordered Semigroups Isometric Representations of Totally Ordered Semigroups

2012-03-25

arxiv.org/abs/1203.5490v1

Mathematics

Operator Algebras

February 21, 2012. Kazan, Russia

Scientific paper

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.

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