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Splittings for C*-correspondences and strong shift equivalence

Kevin Aguyar Brix, Alexander Mundey, Adam Rennie

Abstract

We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to C*-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of C*-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular C*-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant *-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for C*-correspondences.

Splittings for C*-correspondences and strong shift equivalence

Abstract

We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to C*-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of C*-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular C*-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant *-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for C*-correspondences.
Paper Structure (13 sections, 31 theorems, 108 equations)

This paper contains 13 sections, 31 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Correspondences and Cuntz--Pimsner algebras
  3. C*-modules and correspondences
  4. Frames
  5. Topological graphs
  6. Strong shift equivalence
  7. In-splits
  8. In-splits for topological graphs
  9. Noncommutative in-splits
  10. In-splits and diagonal-preserving isomorphism
  11. Out-splits
  12. Out-splits for directed graphs
  13. Noncommutative out-splits

Key Result

Lemma 2.6

Let $(\alpha,\beta)\colon (\phi_X, {}_AX_A) \to (\phi_Y, {}_BY_B)$ be a covariant correspondence morphism, and let $(\iota_A, \iota_X)$ and $(\iota_B,\iota_Y)$ be universal covariant representations of $\mathcal{O}_X$ and $\mathcal{O}_Y$, respectively. Then there is an induced gauge-equivariant $*$- If $\alpha$ is injective, then $\alpha\times \beta$ is injective.

Theorems & Definitions (103)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7
  • proof
  • Definition 2.8
  • Definition 2.9
  • ...and 93 more