$k$-graph algebras are iterated Cuntz-Pimsner algebras -- from the bottom up

TL;DR

This paper presents a new bottom-up method to express $k$-graph $C^*$-algebras as iterated Cuntz-Pimsner algebras, starting from graph algebras and working upward through product systems.

Contribution

It introduces a novel iterative approach based on Pimsner's theorem, contrasting with previous methods that used linking algebras and $(k-1)$-graph algebras.

Findings

Successfully expresses $k$-graph $C^*$-algebras as iterated Cuntz-Pimsner algebras.

Provides a functorial perspective via decategorization of recent theorems.

Demonstrates the approach on product systems over $ ats^k$.

Abstract

We introduce a new method of expressing a $k$-graph $C^*$-algebra as a Cuntz-Pimsner algebra. Kumjian, Pask, and Sims have done this directly, using a linking algebra approach and a $(k-1)$-graph algebra. This can be iterated downward. Our process, on the other hand, starts at the bottom, with Pimsner's theorem for graph algebras, and iterates upward. We actually work with product systems over $\mathbb N^k$, and the result for $k$-graphs is a special case. Our iteration step involves a ``decategorization'' of a recent theorem showing that the Cuntz-Pimsner construction is functorial at the level of ``enchilada categories''.