Relative Morita equivalence of Cuntz--Krieger algebras and flow equivalence of topological Markov shifts

Abstract

In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger algebras with their canonical masas, they are relatively Morita equivalent if and only if their underlying two-sided topological Markov shifts $(\bar{X}_A,\barσ_A)$ and $(\bar{X}_B,\barσ_B)$ are flow equivalent. We also introduce a relative version of the Picard group ${\operatorname{Pic}}(\mathcal{A}, \mathcal{D})$ for the pair $(\mathcal{A}, \mathcal{D})$ of $C^*$-algebras and study them for the Cuntz--Krieger pair $(\mathcal{O}_A, \mathcal{D}_A)$.