| Title:
|
Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^*$-algebras (English) |
| Author:
|
Kodaka, Kazunori |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
4 |
| Year:
|
2022 |
| Pages:
|
435-460 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 (English) |
| Keyword:
|
$C^*$-algebraic bundle |
| Keyword:
|
equivalence bundle |
| Keyword:
|
inclusions of $C^*$-algebra |
| Keyword:
|
strong Morita equivalence |
| MSC:
|
46L05 |
| MSC:
|
46L08 |
| idZBL:
|
Zbl 07655819 |
| idMR:
|
MR4512166 |
| DOI:
|
10.21136/MB.2021.0005-21 |
| . |
| Date available:
|
2022-11-16T11:14:51Z |
| Last updated:
|
2023-04-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151090 |
| . |
| Reference:
|
[1] Abadie, F., Ferraro, D.: Equivalence of Fell bundles over groups.J. Oper. Theory 81 (2019), 273-319. Zbl 1438.46066, MR 3959060, 10.7900/jot.2018feb02.2211 |
| Reference:
|
[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. Zbl 0362.46043, MR 0463928, 10.2140/pjm.1977.71.349 |
| Reference:
|
[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. Zbl 0846.46031, MR 1304338, 10.4153/CJM-1994-065-5 |
| Reference:
|
[4] Jensen, K. K., Thomsen, K.: Elements of $KK$-Theory.Mathematics: Theory & Applications. Birkhäuser, Basel (1991). Zbl 1155.19300, MR 1124848, 10.1007/978-1-4612-0449-7 |
| Reference:
|
[5] Kajiwara, T., Watatani, Y.: Crossed products of Hilbert $C^*$-bimodules by countable discrete groups.Proc. Am. Math. Soc. 126 (1998), 841-851. Zbl 0890.46047, MR 1423344, 10.1090/S0002-9939-98-04118-5 |
| Reference:
|
[6] Kajiwara, T., Watatani, Y.: Jones index theory by Hilbert $C^*$-bimodules and $K$-Theory.Trans. Am. Math. Soc. 352 (2000), 3429-3472. Zbl 0954.46034, MR 1624182, 10.1090/S0002-9947-00-02392-8 |
| Reference:
|
[7] Kodaka, K.: The Picard groups for unital inclusions of unital $C^*$-algebras.Acta Sci. Math. 86 (2020), 183-207. Zbl 1463.46081, MR 4103011, 10.14232/actasm-019-271-1 |
| Reference:
|
[8] Kodaka, K., Teruya, T.: Involutive equivalence bimodules and inclusions of $C^*$-algebras with Watatani index 2.J. Oper. Theory 57 (2007), 3-18. Zbl 1113.46057, MR 2301934 |
| Reference:
|
[9] Kodaka, K., Teruya, T.: A characterization of saturated $C^*$-algebraic bundles over finite groups.J. Aust. Math. Soc. 88 (2010), 363-383. Zbl 1204.46032, MR 2827423, 10.1017/S1446788709000445 |
| Reference:
|
[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. Zbl 1402.46039, MR 3820258, 10.1017/S1446788717000301 |
| Reference:
|
[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. Zbl 07282487, MR 4165509, 10.4064/sm190424-5-1 |
| Reference:
|
[12] Rieffel, M. A.: $C^*$-algebras associated with irrational rotations.Pac. J. Math. 93 (1981), 415-429. Zbl 0499.46039, MR 0623572, 10.2140/pjm.1981.93.415 |
| Reference:
|
[13] Watatani, Y.: Index for $C^*$-subalgebras.Mem. Am. Math. Soc. 424 (1990), 117 pages. Zbl 0697.46024, MR 0996807, 10.1090/memo/0424 |
| . |
