Representations of higher-rank graph $C^*$-algebras associated to $\Lambda$-semibranching function systems

TL;DR

This paper develops a new method for constructing faithful, separable representations of $C^*$-algebras associated with higher-rank graphs, providing a new characterization of $ ext{Lambda}$-semibranching systems and numerous examples.

Contribution

It introduces an alternative characterization of $ ext{Lambda}$-semibranching systems that simplifies verifying conditions and enables the construction of faithful separable representations for a broad class of higher-rank graph $C^*$-algebras.

Findings

New characterization of $ ext{Lambda}$-semibranching systems

Construction of faithful separable representations for all row-finite source-free $k$-graphs

Provision of new examples satisfying the characterization

Abstract

In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $\Lambda$. We begin by giving an alternative characterization of the $\Lambda$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.