Coactions of a finite-dimensional $$C^*$$-Hopf algebra on unital $$C^*$$-algebras, unital inclusions of unital $$C^*$$-algebras and strong Morita equivalence.(English)Zbl 1466.46045

Summary: Let $$A$$ and $$B$$ be unital $$C^*$$-algebras and let $$H$$ be a finite-dimensional $$C^*$$-Hopf algebra. Let $$H^0$$ be its dual $$C^*$$-Hopf algebra. Let $$(\rho,u)$$ and $$(\sigma,v)$$ be twisted coactions of $$H^0$$ on $$A$$ and $$B$$, respectively. In this paper, we show the following theorem: Suppose that the unital inclusions $$A\subset A\rtimes_{\rho,u}H$$ and $$B\subset B\rtimes_{\sigma,v}H$$ are strongly Morita equivalent. If $$A'\cap(A\rtimes_{\rho,u}H)=\mathbb{C}1$$, then there is a $$C^*$$-Hopf algebra automorphism $$\lambda^0$$ of $$H^0$$ such that the twisted coaction $$(\rho,u)$$ is strongly Morita equivalent to the twisted coaction $$((\mathrm{id}_B\otimes\lambda^0)\circ\sigma, (\mathrm{id}_B\otimes\lambda^0 \otimes\lambda^0)(v))$$ induced by $$(\sigma,v)$$ and $$\lambda^0$$.

MSC:

 46L05 General theory of $$C^*$$-algebras 46L08 $$C^*$$-modules
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References:

 [1] S. Baaj et G. Skandalis,C∗-algèbres de Hopf et théorie de Kasparov équivariante, K-Theory 2 (1989), 683-721. · Zbl 0683.46048 [2] S. Baaj et G. Skandalis,Unitaires multiplicatifs et dualité pour les produits croisés de C∗-algèbres, Ann. Sci. École Norm. Sup. 26 (1993), 425-488. · Zbl 0804.46078 [3] R. J. Blattner, M. Cohen and S. Montgomery,Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986), 671-711. · Zbl 0619.16004 [4] M. Z. Guo and X. X. Zhang,Takesaki-Takai duality theorem in HilbertC∗-modules, Acta Math. Sinica (English Ser.) 20 (2004), 1079-1088. · Zbl 1099.46039 [5] M. Izumi,Subalgebras of infiniteC∗-algebras with finite Watatani indices. II. Cuntz- Krieger algebras, Duke Math. J. 91 (1998), 409-461. · Zbl 0949.46023 [6] M. Izumi,Inclusions of simpleC∗-algebras, J. Reine Angew. Math. 547 (2002), 97- 138. · Zbl 1007.46048 [7] T. Kajiwara and Y. Watatani,Jones index theory by HilbertC∗-bimodules and Ktheory, Trans. Amer. Math. Soc. 352 (2000), 3429-3472. · Zbl 0954.46034 [8] K. Kodaka,Equivariant Picard groups ofC∗-algebras with finite dimensionalC∗-Hopf algebra coactions, Rocky Mountain J. Math. 47 (2017), 1565-1615. · Zbl 1387.46044 [9] K. Kodaka and T. Teruya,Inclusions of unitalC∗-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensionalC∗-Hopf algebras, Math. Scand. 104 (2009), 221-248. · Zbl 1169.46034 [10] K. Kodaka and T. Teruya,The Rohlin property for coactions of finite dimensional C∗-Hopf algebras on unitalC∗-algebras, J. Operator Theory 74 (2015), 329-369. · Zbl 1389.46052 [11] K. Kodaka and T. Teruya,The strong Morita equivalence for coactions of a finitedimensionalC∗-Hopf algebra on unitalC∗-algebras, Studia Math. 228 (2015), 259- 294. · Zbl 1351.46054 [12] K. Kodaka and T. Teruya,The strong Morita equivalence for inclusions ofC∗-algebras and conditional expectations for equivalence bimodules, J. Austral. Math. Soc. 105 (2018), 103-144. · Zbl 1402.46039 [13] M. A. Rieffel,C∗-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. · Zbl 0499.46039 [14] M. E. Sweedler,Hopf Algebras, Benjamin, New York, 1969. · Zbl 0194.32901 [15] W. Szymański,Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994), 519-528. · Zbl 0802.46076 [16] W. Szymański and C. Peligrad,Saturated actions of finite dimensional Hopf∗-algebras onC∗-algebras, Math. Scand. 75 (1994), 217-239. · Zbl 0854.46054 [17] Y. Watatani,Index forC∗-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117 pp.
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