Monic representations of finite higher-rank graphs

TL;DR

This paper introduces and analyzes monic representations of $C^*$-algebras associated with finite higher-rank graphs, establishing their properties and connections to existing representation models.

Contribution

It defines monic representations for these algebras, links them to $ ext{semibranching}$ representations, and provides a universal model for nonnegative cases.

Findings

Monic representations admit a cyclic vector on the infinite path space.

A connection is established between monic and $ ext{semibranching}$ representations.

A universal model for nonnegative monic representations is constructed.

Abstract

In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $C^*$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $\Lambda$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.