Topological boundaries of covariant representations

TL;DR

This paper introduces a boundary associated with covariant representations of C*-dynamical systems, analyzing group actions on this boundary and linking trace rigidity to the group's action faithfulness.

Contribution

It defines a new boundary for covariant representations and explores its properties, connecting group action amenability and trace rigidity in C*-algebras.

Findings

The boundary $_{}$ is well-defined for covariant representations.

Group actions on the boundary relate to amenability properties.

Trace rigidity correlates with the faithfulness of the group action.

Abstract

We associate a boundary $\mathcal B_{\pi,u}$ to each covariant representation $(\pi,u,H)$ of $C^*$-dynamical system $(G,A,\alpha)$ and study the action of $G$ on $\mathcal B_{\pi,u}$ and its amenability properties. We relate rigidity properties of traces on the associated crossed product C*-algebra to faithfulness of action of the group on this boundary.