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Shift equivalences through the lens of Cuntz-Krieger algebras

Toke Meier Carlsen, Adam Dor-On, Søren Eilers

Abstract

Motivated by Williams' problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE. Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure a delicate difference between SSE and SE in terms of Pimsner dilations for C*-correspondences of adjacency matrices, and use this distinction to refute a proof from a previous paper.

Shift equivalences through the lens of Cuntz-Krieger algebras

Abstract

Motivated by Williams' problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE. Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure a delicate difference between SSE and SE in terms of Pimsner dilations for C*-correspondences of adjacency matrices, and use this distinction to refute a proof from a previous paper.

Paper Structure

This paper contains 7 sections, 20 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shift equivalence and Cuntz-Pimsner algebras
  4. Compatible shift equivalence
  5. Representable shift equivalence
  6. Strong Morita shift equivalence
  7. Shift equivalences through the lens

Key Result

Theorem 1.3

Suppose $A$ and $B$ are two finite essential matrices with entries in $\mathbb{N}$. Then the following are equivalent,

Theorems & Definitions (60)

  • Definition 1.1: Wil73
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 50 more