$\mathbb{N}$-Graph $C^*$-Algebras

TL;DR

This paper extends the concept of $k$-graph $C^*$-algebras to infinite rank $bN$-graphs, establishing foundational theorems, ideal structures, and algebraic parallels for these generalized algebras.

Contribution

It introduces $bN$-graph $C^*$-algebras, generalizing $k$-graph algebras to infinite rank, and proves key theorems and structural properties.

Findings

Analogues of Gauge Invariant Uniqueness and Cuntz-Krieger Theorems established.

$bN$-graph $C^*$-algebras are inductive limits of $k$-graph $C^*$-algebras.

Described gauge-invariant ideal structure and vertex-set for regular ideals.

Abstract

In this paper we generalize the notion of a $k$-graph into (countable) infinite rank. We then define our $C^*$-algebra in a similar way as in $k$-graph $C^*$-algebras. With this construction we are able to find analogues to the Gauge Invariant Uniqueness and Cuntz-Krieger Uniqueness Theorems. We also show that the $\mathbb{N}$-graph $C^*$-algebras can be viewed as the inductive limit of $k$-graph $C^*$-algebras. This gives a nice way to describe the gauge-invariant ideal structure. Additionally, we describe the vertex-set for regular gauge-invariant ideals of our $N$-graph $C^*$-algebras. We then take our construction of the $\mathbb{N}$-graph into the algebraic setting and receive many similarities to the $C^*$-algebra construction.