# Journal of Operator Theory

Volume 74, Issue 1, Summer 2015 pp. 23-43.

Operator algebras and representations from commuting
semigroup actions

**Authors**:
Benton L. Duncan (1) and Justin R. Peters (2)

**Author institution:**(1) Department of Mathematics, North Dakota State
University, Fargo, North Dakota, U.S.A.

(2) Department of Mathematics, Iowa State University, Ames, Iowa, U.S.A.

**Summary: **Let $\mathcal{S}$ 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.
The $C^*$-envelope of the tensor algebra is 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.

**DOI: **http://dx.doi.org/10.7900/jot.2014apr16.2027

**Keywords: **semigroup dynamical system, $C^*$-envelope,
tensor algebra, semicrossed product, Shilov representation

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