Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^*$-algebras

Abstract

Let $\mathcal{A}= \{A_t \}_{t \in G}$ and $\mathcal{B}= \{B_t \}_{t\in G}$ be $C^*$-algebraic bundles over a finite group $G$. Let $C=\oplus_{t \in G}A_t$ and $D=\oplus_{t\in G}B_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we shall show that if there is an equivalence $\mathcal{A}-\mathcal{B}$-bundle over $G$ with some properties, then the unital inclusions of unital $C^*$-algebras $A \subset C$ and $B \subset D$ induced by $\mathcal{A}$ and $\mathcal{B}$ are strongly Morita equivalent. Also, we suppose that $\mathcal{A}$ and $\mathcal{B}$ are saturated and that $A' \cap C= \mathbf{C} 1$. We shall show that if $A \subset C$ and $B \subset D$ are strongly Morita equivalent, then there are an automorphism $f$ of $G$ and an equivalence bundle $\mathcal{A}-\mathcal{B}^f $-bundle over $G$ with the some properties, where $\mathcal{B}^f$ is the $C^*$-algebraic bundle induced by $\mathcal{B}$ and $f$, which is defined by $\mathcal{B}^f = \{B_{f(t)} \}_{t \in G}$. Furthermore, we shall give an application.