Operator algebras and representations from commuting semigroup actions

TL;DR

This paper investigates operator algebras arising from commuting semigroup actions on compact spaces, characterizing their C*-envelopes and analyzing natural classes of representations.

Contribution

It provides two characterizations of the C*-envelope of the tensor algebra and studies natural representations like regular and orbit representations.

Findings

C*-envelope characterized via Katsura's approach and projective limits

C*-envelope expressed as a crossed product C*-algebra

Analysis of regular and orbit representations with Shilov properties

Abstract

Let $\sS$ be a countable, abelian semigroup of continuous surjections on a compact metric space $X$. Corresponding to this dynamical system we associate two operator algebras, the tensor algebra, and the semicrossed product. There is a unique smallest C$^*$-algebra into which an operator algebra is completely isometrically embedded, which is the C$^*$-envelope. We provide two distinct characterizations of the C$^*$-envelope of the tensor algebra; one developed in a general setting by Katsura, and the other using tools of projective and inductive limits, which gives the C$^*$-envelope as a crossed product C$^*$-algebra. We also study two natural classes of representations, the left regular representations and the orbit representations. The first is Shilov, and the second has a Shilov resolution.