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

arXiv2202.08327MaRDI QIDQ6391345

Tim Schenkel

Publication date: 16 February 2022

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 $mathbbN$-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 $mathbbN$-graph into the algebraic setting and receive many similarities to the $C^{*}$-algebra construction.

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