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

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 -graph into (countable) infinite rank. We then define our -algebra in a similar way as in -graph -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 -graph -algebras can be viewed as the inductive limit of -graph -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 -graph -algebras. We then take our construction of the -graph into the algebraic setting and receive many similarities to the -algebra construction.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Noncommutative and Quantum Gravity Theories · Mathematical Analysis and Transform Methods
