Coactions and Skew Products for Topological Quivers

TL;DR

This paper explores the construction of skew product topological quivers from cocycles and establishes their relationship with coactions on C*-algebras, revealing Morita equivalence results.

Contribution

It introduces a method to construct skew product quivers from cocycles and links their C*-algebras via coactions and crossed products.

Findings

Skew product quivers can be constructed from cocycles on topological quivers.

The C*-algebra of a skew product is isomorphic to a crossed product by a coaction.

Reduced crossed products by dual actions are Morita equivalent to original quiver C*-algebras.

Abstract

Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver, and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the C*-algebra of the skew product and a certain native coaction on the C*-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the C*-algebra of the original quiver.