Operator algebras and representations from commuting semigroup actions

Details Operator algebras and representations from commuting semigroup actions Operator algebras and representations from commuting semigroup actions

2010-08-13

arxiv.org/abs/1008.2244v2

Mathematics

Operator Algebras

25 pages There are changes in the section on orbit representations; a new example is included

Scientific paper

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.

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