Operator algebras and representations from commuting semigroup actions

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${}^{\ast }$-algebra into which an operator algebra is completely isometrically embedded, which is the C${}^{\ast }$-envelope. We provide two distinct characterizations of the C${}^{\ast }$-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${}^{\ast }$-envelope as a crossed product C${}^{\ast }$-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.

25 pages This version has many minor corrections throughout