Group Actions on Product Systems

TL;DR

This paper introduces the crossed product construction for product systems under group actions, proving preservation of properties like row-finiteness and faithfulness for amenable groups, with applications to $k$-graphs and higher rank algebras.

Contribution

It defines the crossed product of a product system by a group and proves key properties are preserved under amenable group actions.

Findings

Crossed product of a row-finite, faithful product system remains row-finite and faithful.

Examples demonstrate applications to group actions on $k$-graphs.

Applications extend to higher rank Doplicher-Roberts algebras.

Abstract

We introduce the concept of crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We illustrate with examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.