Classification of representations of higher-rank graph C*-algebras

TL;DR

This paper introduces new methods for constructing and classifying representations of higher-rank graph C*-algebras, including a novel dimension vector and a spectral parametrization.

Contribution

It develops a functorial, explicit approach using a non-self-adjoint algebra's representation theory and a new dimension vector for classification.

Findings

Introduced a new dimension vector for representations.

Constructed a manifold parametrizing spectral components.

Provided explicit, functorial classification techniques.

Abstract

We develop new techniques for the construction and classification of representations of row-finite and locally convex higher-rank graph C*-algebras O. This class includes Cuntz--Krieger algebras associated to row-finite directed graphs. Our approach relies on the representation theory of a certain non-self-adjoint algebra and a lifting process of representations. We introduce a novel dimension vector for representations of O yielding a countable partition of the spectrum. Given a Cuntz--Krieger algebra and a finite dimension vector, we construct a smooth manifold parametrising the corresponding spectral component. Our techniques are both explicit and functorial.