Partial dynamical systems and C*-algebras generated by partial isometries

TL;DR

This paper explores the structure of C*-algebras generated by partial isometries, focusing on their representation theory, simplicity, and ideal structure through the lens of partial dynamical systems and partial actions.

Contribution

It provides a unified framework to analyze C*-algebras from partial isometries using partial actions, including characterizations of faithfulness, simplicity, and ideal structure.

Findings

Characterization of faithful representations of the C*-algebras

Conditions for simplicity of the crossed product structures

Explicit analysis of C*-algebras from partial isometries in various contexts

Abstract

A collection of partial isometries whose range and initial projections satisfy a specified set of conditions often gives rise to a partial representation of a group. The C*-algebra generated by the partial isometries is thus a quotient of the universal C*-algebra for partial representations of the group, from which it inherits a crossed product structure, of an abelian C*-algebra by a partial action of the group. Questions of faithfulness of representations, simplicity, and ideal structure of these C*-algebras can then be addressed in a unified manner from within the theory of partial actions. We do this here, focusing on two key properties of partial dynamical systems, namely amenability and topological freeness; they are the essential ingredients of our main results in which we characterize faithful representations, simplicity and the ideal structure of crossed products. As applications we consider three situations involving C*-algebras generated by partial isometries: partial representations of groups, Toeplitz algebras of quasi-lattice ordered groups, and Cuntz-Krieger algebras. These C*-algebras share a crossed product structure which we give here explicitly and which we use to study them in terms of the underlying partial actions.