Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T00:18:11.006Z Has data issue: false hasContentIssue false

The Operator Biprojectivity of the Fourier Algebra

Published online by Cambridge University Press:  20 November 2018

Peter J. Wood*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Arsac, G., Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire. Publ. Dép. Math. (Lyon) 13 (1976), 1101.Google Scholar
[2] Blecher, D. P., The standard dual of an operator space. Pacific J. Math. 153 (1992), 1530.Google Scholar
[3] Blecher, D. P. and Paulsen, V. I., Tensor products of operator spaces. J. Funct. Anal. 99 (1991), 262292.Google Scholar
[4] Cartan, H. and Eilenberg, S., Homological Algebra. Princeton University Press, Princeton, 1956.Google Scholar
[5] Curtis, P. C. and Loy, R. J., Amenable Banach algebras. J. London Math Soc. 40 (1989), 89104.Google Scholar
[6] Effros, E. G. and Ruan, Z.-J., On the abstract characterization of operator spaces. Proc. Amer.Math. Soc. 119 (1990), 579584.Google Scholar
[7] Effros, E. G. and Ruan, Z.-J., On approximation properties for operator spaces. Internat. J. Math. 1 (1990), 163187.Google Scholar
[8] Effros, E. G. and Ruan, Z.-J., Operator convolution algebras: an approach to quantum groups. Preprint.Google Scholar
[9] Eymard, P., L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181236.Google Scholar
[10] Forrest, B., Amenability and bounded approximate identities in ideals of A(G). Illinois J. Math. (1) 34 (1990), 125.Google Scholar
[11] Forrest, B. and Wood, P. J., Cohomology and the operator space structure of the Fourier algebra and its second dual. Indiana Univ.Math. J. (3) 50 (2001), 12171240.Google Scholar
[12] Johnson, B., Cohomology in Banach algebras. Mem. Amer.Math. Soc. 127, 1972.Google Scholar
[13] Khelemskii, A. Ya., Flat Banach modules and amenable algebras. Trans.Moscow Math. Soc. 47(1984) 179218, 247; Amer. Math. Soc. Transl. (1985) 199–224.Google Scholar
[14] Khelemskii, A. Ya., The Homology of Banach and Topological Algebras. Mathematics and its Applications (Soviet Series) 41, Kluwer Academic Publishers, 1989.Google Scholar
[15] Khelemskii, A. Ya., Banach and Locally Convex Algebras. Oxford Science Publications, 1993.Google Scholar
[16] Kumar, A. and Sinclair, A. M., Equivalence of norms on operator space tensor products of C*algebras. Trans. Amer. Math. Soc. (5) 350 (1998), 20332048.Google Scholar
[17] Losert, V., Properties of groups without the property P1. Comment.Math. Helv. (3) 54 (1979), 133139.Google Scholar
[18] MacLane, S., Homology. Springer-Verlag, 1963.Google Scholar
[19] Ruan, Z.-J., The operator amenability of A(G). Amer. J. Math. 117 (1995), 14491474.Google Scholar
[20] Ruan, Z.-J. and Xu, G., Splitting properties of operator bimodules and the operator amenability of Kac algebras. In: 16th Operator Theory Conference Proceedings, Theta Found. Bucharest, 1997, 193–216.Google Scholar
[21] Steiniger, H., Finite-dimensional extensions of Fourier algebras. Preprint.Google Scholar
[22] Taylor, J. L., Homology and cohomology for topological algebras. Advances in Math. 9 (1972), 137182.Google Scholar
[23] Taylor, K., Geometry of the Fourier algebras and locally compact groups with atomic unitary representations. Math. Ann. (2) 262 (1983), 183190.Google Scholar
[24] Wood, P. J., Complemented ideals in the Fourier algebra of a locally compact group. Proc. Amer.Math. Soc. (2) 128 (2000), 445451.Google Scholar