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Tensor Products of Operator Spaces II

Published online by Cambridge University Press:  20 November 2018

David P. Blecher
David P. Blecher*
Affiliation:
Mathematics Department, University of Houston, Houston, Texas U.S.A. 77204-3476
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Abstract

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Together with Vern Paulsen we were able to show that the elementary theory of tensor norms of Banach spaces carries over to operator spaces. We suggested that the Grothendieck tensor norm program, which was of course enormously important in the development of Banach space theory, be carried out for operator spaces. Some of this has been done by the authors mentioned above, and by Effros and Ruan. We give alternative developments of some of this work, and otherwise continue the tensor norm program.

Keywords

46M0547D1546C1047D35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 1 , 01 December 1991 , pp. 75 - 90
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Arveson, W.B., Subalgebras of C-algebras I, Acta Math. 123(1969), 142–224.Google Scholar
2. Blecher, D.P., Geometry of the tensor product of C* -algebras, Math. Proc. Camb. Philos. Soc. 104(1988), 119–127.Google Scholar
3. Blecher, D.P., Tensor products which do not preserve operator algebras, Math. Proc. Camb. Philos. Soc. 108(1990), 395–403.Google Scholar
4. Blecher, D.P. , Commutativity in operator algebras, Proc. Amer. Math. Soc. 109(1990), 709–715.Google Scholar
5. Blecher, D.P., The standard dual of an operator space, Pacific J.Math. (1989), to appear.Google Scholar
6. Blecher, D.P. and Paulsen, V.I., Tensor products of operator spaces, J. Funct. Anal. 99(1991), 262–292.Google Scholar
7. Blecher, D.P., Ruan, Z-J. and Sinclair, A.M., A characterization of operator algebras, Funct J. Anal. 89(1990), 188–201.Google Scholar
8. Carne, T.K.. Representation theory for tensor products of Banach algebras, Math. Proc. Camb. Philos. Soc. 90(1981), 445–362.Google Scholar
9. Christensen, E. and Sinclair, A.M., Representations of completely bounded multilinear operators, J. Funct. Anal. 72(1987), 151–181.Google Scholar
10. Effros, E.G. and Exel, R., On multilinear double commutant theorems. Operator Algebras and Applications Vol. 1, London Math. Soc. Lecture Notes Series 135, 1988.Google Scholar
11. Effros, E.G. and Kishimoto, A., Module maps and Hochschild-Johnson cohomology, Indiana Math. J. 37(1989), 257–276.Google Scholar
12. Effros, E.G. and Ruan, Z-J., On matricially normed spaces, Pacific J. Math. 132( 1988), 243–264.Google Scholar
13. Effros, E.G., A new approach to operator spaces, preprint, 1989.Google Scholar
14. Effros, E.G., On approximation properties for operator spaces, International J. Math. 1(1990), 163–187.Google Scholar
15. Effros, E.G., Self-duality for the Haagerup tensor product and Hilbert space factorization, preprint, 1990.Google Scholar
16. Gilbert, J. and Leigh, T.J., Factorization, tensor products and bilinear forms in Banach space theory. Notes in Banach spaces, University of Texas Press, Austin and London, 1980. 182–305.Google Scholar
17. Grothendieck, A., Résumé de la théorie métrique des produits tensoriels topologiques, Bull. Soc. Mat. Sao Paulo 8(1956), 1–79.Google Scholar
18. Haagerup, U., Injectivity and decompositions of completely bounded maps. Lecture Notes in Math. 1132, Springer-Verlag, Berlin, 1983. 170–222.Google Scholar
19. John, F., Extremum problems with inequalities as subsidiary conditions. Courant Anniversary Volume, New York, Interscience, 1948.Google Scholar
20. Kraus, J., The slice map problem for a-weakly closed subspaces of von Neumann algebras, Trans. Amer. Math. Soc. 279(1983), 357–376.Google Scholar
21. Paulsen, V.I., Completely bounded maps and dilations. Pitman Research Notes in Math., Longman, London, 1986.Google Scholar
22. Paulsen, V.I., Representations of function algebras, abstract operator spaces, and Banach space geometry, preprint, 1990.Google Scholar
23. Paulsen, V I. and Smith, R.R., Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73(1987), 258–276.Google Scholar
24. Pisier, G., Grothendieck's theorem for non-commutative C*-algebras with an appendix on Grothendieck's constants, J. Funct. Anal. 29(1978), 397–415.Google Scholar
25. Pisier, G., Factorization of linear operators and the geometry of Banach space. C.B. Series M.S. 60, Amer. Math. Soc. Publ., 1986.Google Scholar
26. Pisier, G., Factorization of operator valued analytic functions, preprint, 1989.Google Scholar
27. Ruan, Z-J., Subspaces of C*-algebras, J. Funct. Anal. 76(1988), 217–230.Google Scholar
28. Schatten, R., A theory of cross spaces. Anals. of Math. Studies 26, Princeton, 1950.Google Scholar
29. Smith, R.R., Completely bounded module maps and the Haagerup tensor product, preprint, 1990.Google Scholar
30. Takesaki, M., Theory of operator algebras I. Springer-Verlag, Berlin, 1979.Google Scholar
31. Tomiyama, J., Tensor products and projections of norm one in von Neumann algebras. Lecture notes, University of Copenhagen, 1970.Google Scholar
32. Haagerup, U., The a-tensor product for C*-algebras, unpublished manuscript (1980)..Google Scholar