Normal Extensions of Representations of Abelian Semigroups

TL;DR

This paper investigates conditions under which representations of abelian semigroups can be extended to normal representations, focusing on extending generators to commuting normal operators and generalizing existing results to lattice-ordered structures.

Contribution

It establishes that extending generators to commuting normal operators suffices for extending the entire representation and generalizes Athavale's result to abelian lattice-ordered semigroups.

Findings

Extending generators to commuting normals suffices for full extension.

A generalization of Athavale's result to abelian lattice-ordered semigroups.

Not all commuting subnormal operators have commuting normal extensions.

Abstract

A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation of an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extended a result due to Athavale to representations on abelian lattice ordered semigroups.