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Keywords:
$C^*$-algebraic bundle; equivalence bundle; inclusions of $C^*$-algebra; strong Morita equivalence
$C^*$-algebraic bundle; equivalence bundle; inclusions of $C^*$-algebra; strong Morita equivalence
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 -2
References:
[1] Abadie, F., Ferraro, D.: Equivalence of Fell bundles over groups. J. Oper. Theory 81 (2019), 273-319. DOI 10.7900/jot.2018feb02.2211 | MR 3959060 | Zbl 1438.46066
[2] Brown, L. G., Green, P., Rieffel, M. A.: Stable isomorphism and strong Morita equivalence of $C^*$-algebras. Pac. J. Math. 71 (1977), 349-363. DOI 10.2140/pjm.1977.71.349 | MR 0463928 | Zbl 0362.46043
[3] Brown, L. G., Mingo, J. A., Shen, N-T.: Quasi-multipliers and embeddings of Hilbert $C^*$-bimodules. Can. J. Math. 46 (1994), 1150-1174. DOI 10.4153/CJM-1994-065-5 | MR 1304338 | Zbl 0846.46031
[4] Jensen, K. K., Thomsen, K.: Elements of $KK$-Theory. Mathematics: Theory & Applications. Birkhäuser, Basel (1991). DOI 10.1007/978-1-4612-0449-7 | MR 1124848 | Zbl 1155.19300
[5] Kajiwara, T., Watatani, Y.: Crossed products of Hilbert $C^*$-bimodules by countable discrete groups. Proc. Am. Math. Soc. 126 (1998), 841-851. DOI 10.1090/S0002-9939-98-04118-5 | MR 1423344 | Zbl 0890.46047
[6] Kajiwara, T., Watatani, Y.: Jones index theory by Hilbert $C^*$-bimodules and $K$-Theory. Trans. Am. Math. Soc. 352 (2000), 3429-3472. DOI 10.1090/S0002-9947-00-02392-8 | MR 1624182 | Zbl 0954.46034
[7] Kodaka, K.: The Picard groups for unital inclusions of unital $C^*$-algebras. Acta Sci. Math. 86 (2020), 183-207. DOI 10.14232/actasm-019-271-1 | MR 4103011 | Zbl 1463.46081
[8] Kodaka, K., Teruya, T.: Involutive equivalence bimodules and inclusions of $C^*$-algebras with Watatani index 2. J. Oper. Theory 57 (2007), 3-18. MR 2301934 | Zbl 1113.46057
[9] Kodaka, K., Teruya, T.: A characterization of saturated $C^*$-algebraic bundles over finite groups. J. Aust. Math. Soc. 88 (2010), 363-383. DOI 10.1017/S1446788709000445 | MR 2827423 | Zbl 1204.46032
[10] Kodaka, K., Teruya, T.: The strong Morita equivalence for inclusions of $C^*$-algebras and conditional expectations for equivalence bimodules. J. Aust. Math. Soc. 105 (2018), 103-144. DOI 10.1017/S1446788717000301 | MR 3820258 | Zbl 1402.46039
[11] Kodaka, K., Teruya, T.: Coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and the strong Morita equivalence. Stud. Math. 256 (2021), 169-185. DOI 10.4064/sm190424-5-1 | MR 4165509 | Zbl 07282487
[12] Rieffel, M. A.: $C^*$-algebras associated with irrational rotations. Pac. J. Math. 93 (1981), 415-429. DOI 10.2140/pjm.1981.93.415 | MR 0623572 | Zbl 0499.46039
[13] Watatani, Y.: Index for $C^*$-subalgebras. Mem. Am. Math. Soc. 424 (1990), 117 pages. DOI 10.1090/memo/0424 | MR 0996807 | Zbl 0697.46024
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