Imprimitivity bimodules of Cuntz--Krieger algebras and strong shift equivalences of matrices

TL;DR

This paper establishes a link between matrix equivalences and Morita equivalences of Cuntz--Krieger algebras, using a basis-related notion of Morita equivalence to characterize elementary matrix equivalence.

Contribution

It introduces a basis-related Morita equivalence concept for Cuntz--Krieger algebras and proves its equivalence to elementary matrix equivalence.

Findings

Two matrices are elementary equivalent if and only if their Cuntz--Krieger algebras are basis relatedly Morita equivalent.

The paper characterizes matrix equivalence through the structure of associated Cuntz--Krieger algebras.

Establishes a new perspective connecting matrix algebra and operator algebra theory.

Abstract

In this paper, we will introduce a notion of basis related Morita equivalence in the Cuntz--Krieger algebras $({{\mathcal{O}}_A}, \{S_a\}_{a \in E_A})$ with the canonical right finite basis $\{S_a\}_{a \in E_A}$ as Hilbert $C^*$-bimodule, and prove that two nonnegative irreducible matrices $A$ and $B$ are elementary equivalent, that is, $A = CD, B = DC$ for some nonnegative rectangular matrices $C, D$, if and only if the Cuntz--Krieger algebras $({{\mathcal{O}}_A}, \{S_a\}_{a \in E_A})$ and $({{\mathcal{O}}_B}, \{ S_b\}_{b\in E_B})$ with the canonical right finite bases are basis relatedly elementary Morita equivalent.