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 $C=\bigoplus_{t\in G}A_t$ and $D=\bigoplus_{t\in G}B_t$. 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 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 $A' \cap C= C 1$. We 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 \hbox{$\mathcal{A}-\mathcal{B}^f $}-bundle over $G$ with the above 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 give an application.\looseness-2

Keywords

• $c^*$-algebraic bundle
• equivalence bundle
• inclusions of $c^*$-algebra
• strong morita equivalence