Coverings of skew-products and crossed products by coactions

TL;DR

This paper investigates the structure of coaction crossed products of C*-algebras, especially those arising from skew-products of k-graphs, showing they can be realized as corners of higher-rank graph C*-algebras and relate to topological higher-rank graphs.

Contribution

It demonstrates that coaction crossed products from projective limits of finite groups can be expressed as direct limits and realizes certain crossed products as corners of higher-rank graph C*-algebras, connecting to topological graph theory.

Findings

Crossed product of A by a projective limit coaction is a direct limit of crossed products.

Coaction crossed product of C*(Λ) can be realized as a full corner of a (k+1)-graph C*-algebra.

Connections established with Yeend's topological higher-rank graphs.

Abstract

Consider a projective limit G of finite groups G_n. Fix a compatible family \delta^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction \delta of G on A. We show that the coaction crossed product of A by \delta is isomorphic to a direct limit of the coaction crossed products of A by the \delta^n. If A = C*(\Lambda) for some k-graph \Lambda, and if the coactions \delta^n correspond to skew-products of \Lambda, then we can say more. We prove that the coaction crossed-product of C*(\Lambda) by \delta may be realised as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend's topological higher-rank graphs and their C*-algebras.