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

Valentin Deaconu, Menevşe Eryüzlü Paulovicks, S. Kaliszewski, John Quigg

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''.

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

Abstract

We introduce a new method of expressing a -graph -algebra as a Cuntz-Pimsner algebra. Kumjian, Pask, and Sims have done this directly, using a linking algebra approach and a -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 , and the result for -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''.
Paper Structure (6 sections, 17 theorems, 116 equations)

This paper contains 6 sections, 17 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abstract iterative process
  4. Main result
  5. The Structure of the Iterated Product Systems
  6. Proofs of Lemmas

Key Result

Proposition 2.1

For a regular $C^*$-correspondence ${}_{A}X_A$, let $(i_A, i_X): {}_{A}X_A\rightarrow {}_{A}X_A$ be a $C^*$-correspondence automorphism. Then there exists an automorphism $i_{X*}: \mathscr{O} X\rightarrow \mathscr{O} X$ such that

Theorems & Definitions (34)

  • Proposition 2.1
  • Lemma 2.2: dkps
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: fowsim
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 24 more