Topologically irreducible representations of the Banach *-algebra associated with a dynamical system

TL;DR

This paper characterizes infinite-dimensional irreducible representations of a crossed product C*-algebra from a topological dynamical system and examines their restrictions to an associated Banach *-algebra, revealing new ergodic measures.

Contribution

It provides a detailed description of irreducible representations using ergodic measures and introduces a new class of ergodic measures for irrational rotations.

Findings

Restrictions to the  Banach *-algebra are not algebraically irreducible under mild conditions.

A new class of ergodic measures for irrational rotations is constructed.

The description involves ergodic measures and ergodic extensions for tensor products with type I factors.

Abstract

We describe (infinite-dimensional) irreducible representations of the crossed product C$^*$-algebra associated with a topological dynamical system (based on $Z$) and we show that their restrictions to the underling $\ell^1$-Banach $*$-algebra are not algebraically irreducible under mild conditions on the dynamical system. The above description of irreducible representations has two ingredients, ergodic measures on the space and ergodic extensions for the tensor product with type I factors; the latter which may not have been explicitly taken up before will be explored by examples. A new class of ergodic measures is also constructed for irrational rotations on the circle.