Separable representations of higher-rank graphs

TL;DR

This paper provides a comprehensive analysis of separable representations of $C^*$-algebras associated with finite $k$-graphs, introducing new characterizations, examples, and classifications of various representation types.

Contribution

It offers new characterizations of $ ext{Lambda}$-semibranching systems, constructs a faithful separable representation, and fully characterizes monic, purely atomic, and permutative representations of $k$-graph $C^*$-algebras.

Findings

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

Construction of a faithful separable representation for $k$-graphs

Complete classification of monic, atomic, and permutative representations

Abstract

In this monograph we undertake a comprehensive study of separable representations (as well as their unitary equivalence classes) of $C^*$-algebras associated to strongly connected finite $k$-graphs $\Lambda$. We begin with the representations associated to the $\Lambda$-semibranching function systems introduced by Farsi, Gillaspy, Kang, and Packer in \cite{FGKP}, by giving an alternative characterization of these systems which is more easily verified in examples. We present a variety of such examples, one of which we use to construct a new faithful separable representation of any row-finite source-free $k$-graph. Next, we analyze the monic representations of $C^*$-algebras of finite $k$-graphs. We completely characterize these representations, generalizing results of Dutkay and Jorgensen \cite{dutkay-jorgensen-monic} and Bezuglyi and Jorgensen \cite{bezuglyi-jorgensen} for Cuntz and Cuntz-Krieger algebras respectively. We also describe a universal representation for non-negative monic representations of finite, strongly connected $k$-graphs. To conclude, we characterize the purely atomic and permutative representations of $k$-graph $C^*$-algebras, and discuss the relationship between these representations and the classes of representations introduced earlier.