Representations of Higher Rank Graph Algebras

TL;DR

This paper studies representations of higher rank graph algebras, showing how irreducible atomic representations relate to group constructions, characterizing periodicity, and describing the algebra's structure via symmetry subgroups.

Contribution

It introduces a framework linking atomic representations to group construction representations and characterizes the algebra's structure through symmetry subgroups.

Findings

Irreducible atomic *-representations are minimal dilations of group construction representations.

Atomic representations decompose into sums or integrals of these representations.

The algebra's structure is described as a tensor product involving a symmetry subgroup.

Abstract

Let $\Fth$ be a $\Bk$-graph on a single vertex. We show that every irreducible atomic $*$-representation is the minimal $*$-dilation of a group construction representation. It follows that every atomic representation decomposes as a direct sum or integral of such representations. We characterize periodicity of $\Fth$ and identify a symmetry subgroup $H_\theta$ of $\bZ^\Bk$. If this has rank $s$, then $\ca(\Fth) \cong \rC(\bT^s) \otimes \fA$ for some simple C*-algebra $\fA$.