# 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