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 $(\bar{X}_A,\bar{\sigma}_A)$ and $(\bar{X}_B,\bar{\sigma}_B)$ 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 $(\mathcal{O}_A, \mathcal{D}_A)$.