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

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 $C^{\ast }$-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.

20 pages, 6 figures

References in corpus (1)

• Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz--Krieger algebras