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Positive Herz–Schur multipliers and approximation properties of crossed products


For a C*-algebra A and a set X we give a Stinespring-type characterisation of the completely positive Schur A-multipliers on κ(ℓ2(X)) ⊗ A. We then relate them to completely positive Herz–Schur multipliers on C*-algebraic crossed products of the form A α,r G, with G a discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, for A α,r G.

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[A-D] Anantharaman-Delaroche C. Systémes dynamiques non commutatifs et moyennabilité. Math. Ann. 279 (1987), no. 2, 297315.
[BéC] Bédos É. and Conti R.. The Fourier-Stieltjes algebra of a C*-dynamical system. Internat. J. Math. 27 (2016), no. 6, 1650050, 50 pp., available at arXiv:1510.03296.
[Bra] Brannan M. Approximation properties for locally compact quantum groups. Banach Center Publ. 111 (2017), 185232, available at arXiv:1605.01770.
[BrO] Brown N. P. and Ozawa N. C*-algebras and finite dimensional approximations. Amer. Math. Soc. (2008).
[Cho] Choda M. Group factors of the Haagerup type. Proc. Japan Acad. 59, (1983), 174209.
[ChE] Choi M. D. and Effros E. Nuclear C*-algebras and the approximation property. Amer. J. Math. 100 (1978), no. 1, 6179.
[Don] Dong Z. Haagerup property for C*-algebras. J. Math. Anal. Appl. 377, (2010), 631644.
[DoR] Dong Z. and Ruan Z.J. A Hilbert module approach to the Haagerup property. Integral Equations Operator Theory 73 (2012), 431454.
[EfR] Effros E. G. and Ruan Z.–J.. Operator Spaces (Oxford University Press, 2000).
[Gro] Grothendieck A. Résumé de la théorie métrique des produits tensoriels topologiques. Boll. Soc. Mat. Sao-Paulo 8 (1956), 179.
[Jol] Jolissaint P. Haagerup approximation property for finite von Neumann algebras. J. Operator Theory. 48 (2002), 539551.
[Lan] Lance C. On nuclear C*-algebras. J. Funct. Ana. 12 (1973), 157176.
[McK] McKee A. Weak amenability for dynamical systems. Preprint, arXiv:1612.01758.
[MTT] McKee A., Todorov I. G. and Turowska L.. Herz–Schur multipliers of dynamical systems. Preprint, arXiv:1608.01092.
[Men] Meng Q. Haagerup property of C*-algebras. Bull. Aust. Math. Soc. 93 (2016), no. 2, 295300.
[NeS] Neshveyev S. and Størmer E. Dynamical entropy in operator algebras. Series of Modern Surveys in Mathematics. 50 (Springer, 2006).
[Pau] Paulsen V. I. Completely Bounded Maps and Operator Algebras. (Cambridge University Press, 2002).
[Pis] Pisier G. Similarity problems and completely bounded maps. Lecture Notes in Math. 1618 (Springer, 2001).
[SiS] Sinclair A. M. and Smith R. R. The completely bounded approximation property for discrete crossed products. Indiana Univ. Math. J. 46 (1997), no. 4, 13111322.
[You] You C. Haagerup property for C*-crossed products. Bull. Aust. Math. Soc. 95 (2017), no. 1, 144148.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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