Realising the C*-algebra of a higher-rank graph as an Exel crossed product

TL;DR

This paper constructs a C*-algebra from a higher-rank graph using boundary-path spaces, showing it is isomorphic to a Cuntz-Nica-Pimsner algebra, and introduces a new notion of crossed product by semigroups of partial endomorphisms.

Contribution

It introduces a novel approach to realize higher-rank graph C*-algebras as Exel crossed products via boundary-path spaces and compares this with existing definitions.

Findings

C*-algebra of a higher-rank graph is isomorphic to a Cuntz-Nica-Pimsner algebra.

Introduces a new notion of crossed product by semigroups of partial endomorphisms.

Establishes connections with existing crossed product frameworks.

Abstract

We use the boundary-path space of a finitely-aligned k-graph \Lambda to construct a compactly-aligned product system X, and we show that the graph algebra C^*(\Lambda) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.