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Proper actions on imprimitivity bimodules and decompositions of Morita equivalences. (English) Zbl 1031.46068
Summary: We consider a class of proper actions of locally compact groups on imprimitivity bimodules over \(C^*\)-algebras which behave like the proper actions on \(C^*\)-algebras introduced by M. A. Rieffel [Prog. Math. 84, 141-182 (1991; Zbl 0749.22003)]. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel’s theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences, and this gives new information about the amenability of actions on graph algebras.

MSC:
46L08 \(C^*\)-modules
46L05 General theory of \(C^*\)-algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
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References:
[1] Combes, F., Crossed products and Morita equivalence, Proc. London math. soc., 49, 289-306, (1984) · Zbl 0521.46058
[2] Curto, R.E.; Muhly, P.S.; Williams, D.P., Cross products of strongly Morita equivalent C*-algebras, Proc. amer. math. soc., 90, 528-530, (1984) · Zbl 0508.22012
[3] Echterhoff, S.; Raeburn, I., The stabilisation trick for coactions, J. reine angew. math., 470, 181-215, (1996) · Zbl 0929.46054
[4] Echterhoff, S.; Kaliszewski, S.; Quigg, J.; Raeburn, I., Naturality and induced representations, Bull. austral. math. soc., 61, 415-438, (2000) · Zbl 0982.46051
[5] Exel, R., Morita – rieffel equivalence and spectral theory for integrable automorphism groups of C*-algebras, J. funct. anal., 172, 404-465, (2000) · Zbl 0957.46040
[6] A. an Huef, I. Raeburn, Regularity of induced representations and a theorem of Quigg and Spielberg, Math. Proc. Cambridge Philos. Soc., June 2000, to appear (arXiv.math.OA/0009249). · Zbl 1030.46064
[7] an Huef, A.; Raeburn, I.; Williams, D.P., An equivariant Brauer semigroup and the symmetric imprimitivity theorem, Trans. amer. math. soc., 352, 4759-4787, (2000) · Zbl 1001.46044
[8] Kaliszewski, S.; Quigg, J.; Raeburn, I., Skew products and crossed products by coactions, J. operator theory, 46, 411-433, (2001) · Zbl 0996.46028
[9] Kumjian, A.; Pask, D.; Raeburn, I., Cuntz – krieger algebras of directed graphs, Pacific J. math., 184, 161-174, (1998) · Zbl 0917.46056
[10] Meyer, R., Generalized fixed point algebras and square-integrable group actions, J. funct. anal., 186, 167-195, (2001) · Zbl 1003.46036
[11] Pask, D.; Raeburn, I., Symmetric imprimitivity theorems for graph algebras, Internat. J. math., 12, 609-623, (2001) · Zbl 1116.46305
[12] Quigg, J.C.; Spielberg, J., Regularity and hyporegularity in C*-dynamical systems, Houston J. math., 18, 139-152, (1992) · Zbl 0785.46052
[13] Raeburn, I., Induced C*-algebras and a symmetric imprimitivity theorem, Math. ann., 280, 369-387, (1988) · Zbl 0617.22009
[14] I. Raeburn, W. Szymaǹski, Cuntz-Krieger algebras of infinite graphs and matrices, preprint, University of Newcastle, December 1999.
[15] Raeburn, I.; Williams, D.P., Morita equivalence and continuous-trace C*-algebras, Math. surveys and monographs, Vol. 60, (1998), American Mathematical Society Providence, RI · Zbl 0922.46050
[16] Rieffel, M.A., Applications of strong Morita equivalence to transformation group C*-algebras, (), 299-310
[17] Rieffel, M.A., Proper actions of groups on C*-algebras, (), 141-182 · Zbl 0749.22003
[18] M.A. Rieffel, Integrable and proper actions on C*-algebras, and square integrable representations of groups, preprint (arXiv.math.OA/9809098), June 1999.
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