Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group
DOI:
https://doi.org/10.7146/math.scand.a-125997Abstract
We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.
References
Bédos, E., and Conti, R., On discrete twisted $C^*$-dynamical systems, Hilbert $C^*$-modules and regularity, Münster J. Math. 5 (2012), 183-208.
Blackadar, B., Operator algebras. Theory of $C^*$-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry III, Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/3-540-28517-2
Brown, L. G., Green, P., and M. A. Rieffel, M. A., Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363.
Brown, L. G., Mingo, J., and Shen, N. T., Quasi-multipliers and embeddings of Hilbert $C^*$-bimodules, Canad. J. Math. 46 (1994), no. 6, 1150–1174. https://doi.org/10.4153/CJM-1994-065-5
Choda, H., On freely acting automorphisms of operator algebras, Kōdai Math. Sem. Rep. 26 (1974/75), 1-21.
Combes, F., Crossed products and Morita equivalence, Proc. London Math. Soc. (3) 49 (1984), no. 2, 289–306. https://doi.org/10.1112/plms/s3-49.2.289
Curto, R. E., Muhly, P. S., and Williams, D. P., Cross products of strongly Morita equivalent $C^*$-algebras, Proc. Amer. Math. Soc. 90 (1984), no. 4, 528–530. https://doi.org/10.2307/2045024
Gracia-Bondía, J. M., Várilly, J., and Figueroa, H,. Elements of Noncommutative Geometry, Birkhäuser Boston, 2001. https://doi.org/10.1007/978-1-4612-0005-5
Jensen, K. K., and Thomsen, K., Elements of KK-theory, Birkhäuser, 1991. https://doi.org/10.1007/978-1-4612-0449-7
Kodaka, K., The Picard groups for unital inclusions of unital $C^*$-algebras, Acta Sci. Math. (Szged) 86 (2020), no. 1–2, 183-207. https://doi.org/10.14232/actasm-019-271-1
Kodaka, K., and Teruya, T., The strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, Studia Math. 228 (2015), no. 3, 259–294. https://doi.org/10.4064/sm228-3-4
Kodaka, K., and Teruya, T., The strong Morita equivalence for inclusions of $C^*$-algebras and conditional expectations for equivalence bimodules, J. Aust. Math. Soc. 105 (2018), no. 1, 103–144. https://doi.org/10.1017/S1446788717000301
Kodaka, K., and 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, Studia Math. 256 (2021), no. 2, 147–167. https://doi.org/10.4064/sm190424-5-1
Packer, J. A., and Raeburn, I., Twisted crossed products of $C^*$-algebras, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 2, 293-311. https://doi.org/10.1017/S0305004100078129
Pedersen, G. K., $C^* $-algebras and their automorphism groups, Academic Press, London, New York, San Francisco, 1979.
Raeburn, I., and Williams, D. P., Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, 1998. https://doi.org10.1090/surv/060
Zarikian, V., Unique expectations for discrete crossed products, Ann. Funct. Anal. 10 (2019), no.1, 60-71. https://doi.org/10.1215/20088752-2018-0008
