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Shift equivalence relations through the lens of C*-correspondences

Boris Bilich, Adam Dor-On, Efren Ruiz

Abstract

We continue the study of shift equivalence relations from the perspective of C*-bimodule theory. We study emerging shift equivalence relations following work of the second-named author with Carlsen and Eilers, both in terms of adjacency matrices and in terms of their C*-correspondences, and orient them when possible. In particular, we show that if two regular C*-correspondences are strong shift equivalent, then the intermediary C*-correspondences realizing the equivalence may be chosen to be regular. This result provides the final missing piece in answering a question of Muhly, Pask and Tomforde, and is used to confirm a conjecture of Kakariadis and Katsoulis on shift equivalence of C*-correspondences.

Shift equivalence relations through the lens of C*-correspondences

Abstract

We continue the study of shift equivalence relations from the perspective of C*-bimodule theory. We study emerging shift equivalence relations following work of the second-named author with Carlsen and Eilers, both in terms of adjacency matrices and in terms of their C*-correspondences, and orient them when possible. In particular, we show that if two regular C*-correspondences are strong shift equivalent, then the intermediary C*-correspondences realizing the equivalence may be chosen to be regular. This result provides the final missing piece in answering a question of Muhly, Pask and Tomforde, and is used to confirm a conjecture of Kakariadis and Katsoulis on shift equivalence of C*-correspondences.

Paper Structure

This paper contains 5 sections, 26 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Balanced and Aligned shift equivalence for matrices
  4. Strong shift equivalence for C*-correspondences
  5. SSE of graph correspondences implies SSE of matrices

Key Result

Theorem A

The relations on finite essential matrices with entries in $\mathbb{N}$ given by aligned, balanced and compatible shift equivalence of lag $m$ coincide. Consequently, aligned and balanced shift equivalence relations coincide with SSE.

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.3
  • Definition 3.1: CDE
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 48 more