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

@article{Kodaka2022,
abstract = {Let $\mathcal \{A\}=\lbrace A_t \rbrace _\{t\in G\}$ and $\mathcal \{B\}=\lbrace B_t \rbrace _\{t\in G\}$ 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 $A^\{\prime \} \cap C=\{\bf C\} 1$. 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 $\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 $\mathcal \{B\}^f =\lbrace B_\{f(t)\}\rbrace _\{t\in G\}$. Furthermore, we give an application.},
author = {Kodaka, Kazunori},
journal = {Mathematica Bohemica},
keywords = {$C^*$-algebraic bundle; equivalence bundle; inclusions of $C^*$-algebra; strong Morita equivalence},
language = {eng},
number = {4},
pages = {435-460},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^*$-algebras},
url = {http://eudml.org/doc/298628},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Kodaka, Kazunori
TI - Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^*$-algebras
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 4
SP - 435
EP - 460
AB - Let $\mathcal {A}=\lbrace A_t \rbrace _{t\in G}$ and $\mathcal {B}=\lbrace B_t \rbrace _{t\in G}$ 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 $A^{\prime } \cap C={\bf C} 1$. 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 $\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 $\mathcal {B}^f =\lbrace B_{f(t)}\rbrace _{t\in G}$. Furthermore, we give an application.
LA - eng
KW - $C^*$-algebraic bundle; equivalence bundle; inclusions of $C^*$-algebra; strong Morita equivalence
UR - http://eudml.org/doc/298628
ER -