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Positive Herz–Schur multipliers and approximation properties of crossed products

Published online by Cambridge University Press:  14 August 2017

ANDREW MCKEE ,
ADAM SKALSKI ,
IVAN G. TODOROV  and
LYUDMILA TUROWSKA
ANDREW MCKEE
Affiliation:
Mathematical Sciences Research Centre, Queen's University Belfast, Belfast BT7 1NN. e-mail: amckee240@qub.ac.uk
ADAM SKALSKI
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland. e-mail: a.skalski@impan.pl
IVAN G. TODOROV
Affiliation:
Mathematical Sciences Research Centre, Queen's University Belfast, Belfast BT7 1NN. e-mail: i.todorov@qub.ac.uk
LYUDMILA TUROWSKA
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg SE-412 96, Sweden. e-mail: turowska@chalmers.se
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Abstract

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,rG, 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,rG.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 511 - 532
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[A-D] Anantharaman-Delaroche, C. Systémes dynamiques non commutatifs et moyennabilité. Math. Ann. 279 (1987), no. 2, 297–315.Google Scholar
[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.Google Scholar
[Bra] Brannan, M. Approximation properties for locally compact quantum groups. Banach Center Publ. 111 (2017), 185–232, available at arXiv:1605.01770.Google Scholar
[BrO] Brown, N. P. and Ozawa, N. C*-algebras and finite dimensional approximations. Amer. Math. Soc. (2008).Google Scholar
[Cho] Choda, M. Group factors of the Haagerup type. Proc. Japan Acad. 59, (1983), 174–209.Google Scholar
[ChE] Choi, M. D. and Effros, E. Nuclear C*-algebras and the approximation property. Amer. J. Math. 100 (1978), no. 1, 61–79.Google Scholar
[Don] Dong, Z. Haagerup property for C*-algebras. J. Math. Anal. Appl. 377, (2010), 631–644.Google Scholar
[DoR] Dong, Z. and Ruan, Z.J. A Hilbert module approach to the Haagerup property. Integral Equations Operator Theory 73 (2012), 431–454.Google Scholar
[EfR] Effros, E. G. and Ruan, Z.–J.. Operator Spaces (Oxford University Press, 2000).Google Scholar
[Gro] Grothendieck, A. Résumé de la théorie métrique des produits tensoriels topologiques. Boll. Soc. Mat. Sao-Paulo 8 (1956), 1–79.Google Scholar
[Jol] Jolissaint, P. Haagerup approximation property for finite von Neumann algebras. J. Operator Theory. 48 (2002), 539–551.Google Scholar
[Lan] Lance, C. On nuclear C*-algebras. J. Funct. Ana. 12 (1973), 157–176.Google Scholar
[McK] McKee, A. Weak amenability for dynamical systems. Preprint, arXiv:1612.01758.Google Scholar
[MTT] McKee, A., Todorov, I. G. and Turowska, L.. Herz–Schur multipliers of dynamical systems. Preprint, arXiv:1608.01092.Google Scholar
[Men] Meng, Q. Haagerup property of C*-algebras. Bull. Aust. Math. Soc. 93 (2016), no. 2, 295–300.Google Scholar
[NeS] Neshveyev, S. and Størmer, E. Dynamical entropy in operator algebras. Series of Modern Surveys in Mathematics. 50 (Springer, 2006).Google Scholar
[Pau] Paulsen, V. I. Completely Bounded Maps and Operator Algebras. (Cambridge University Press, 2002).Google Scholar
[Pis] Pisier, G. Similarity problems and completely bounded maps. Lecture Notes in Math. 1618 (Springer, 2001).Google Scholar
[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, 1311–1322.Google Scholar
[You] You, C. Haagerup property for C*-crossed products. Bull. Aust. Math. Soc. 95 (2017), no. 1, 144–148.Google Scholar