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

summary: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 $A' \cap C={\bf C} 1$. 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 -

The Rohlin property for inclusions of C∗C^*-algebras with a finite Watatani index

We introduce notions of the Rohlin property and the approximate representability for inclusions of unital $C^*$-algebras. We investigate a dual relation between the Rohlin property and the approximate representability. We prove that a number of classes of unital $C^*$-algebras are closed under inclusions with the Rohlin property, including: AF algebras, AI algebras, AT algebras, and related classes characterized by direct limit decomposition using semiprojective building blocks. $C^*$-algebras with stable rank one. $C^*$-algebras with real rank zero.Comment: We revised the section 4 and its correspondent part in Introduction in the original pape