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 $C^*$-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.