Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-16T22:38:55.228Z Has data issue: false hasContentIssue false
  • English
  • Français

Tensor products which do not preserve operator algebras

Published online by Cambridge University Press:  24 October 2008

David P. Blecher
David P. Blecher
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204–3476, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 395 - 403
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Blecher, D. P.. Geometry of the tensor product of C*-algebras. Ph.D. thesis, University of Edinburgh (1988).Google Scholar
[2]Blecher, D. P.. Geometry of the tensor product of C*-algebras. Math. Proc. Cambridge Philos. Soc. 104 (1988), 119–127.Google Scholar
[3]Blecher, D. P.. Commutativity in operator algebras. Proc. Amer. Math. Soc. (To appear.)Google Scholar
[4]Blecher, P. P. and Paulsen, V. I.. Tensor products of operator spaces. J. Fund. Anal. (To appear.)Google Scholar
[5]Blecher, D. P., Ruan, Z.-J. and Sinclair, A. M.. A characterization of operator algebras. J. Funct. Anal. 89 (1990), 188–201.Google Scholar
[6]Carne, T. K.. Not all H′-algebras are operator algebras. Math. Proc. Cambridge Philos. Soc. 86 (1979), 243–249.Google Scholar
[7]Carne, T. K.. Operator algebras. Exposé No. 15, Seminaire D'Analyse Fonctionelle 1979–80, École Polytechnique, Palaiseau.Google Scholar
[8]Carne, T. K.. Representation theory for tensor products of Banach algebras. Math. Proc. Cambridge Philos. Soc. 90 (1981), 445–463.Google Scholar
[9]Charpentier, Ph.. Q-algebres et produits tensoriels topologiques. These, Orsay (1973).Google Scholar
[10]Christensen, E., Effros, E. G. and Sinclair, A. M.. Completely bounded maps and C*-algebraic cohomology. Invent. Math. 90 (1987), 279–296.CrossRefGoogle Scholar
[11]Christensen, E. and Sinclair, A. M.. Representations of completely bounded multilinear operators. J. Funct. Anal. 72 (1987), 151–181.Google Scholar
[12]Cohn, D. L.. Measure Theory (Birkhauser, 1980).Google Scholar
[13]Dunford, N. and Schwartz, J. T.. Linear Operators, vols 1, 2 and 3 (Wiley, Interscience, 1958, 1963, 1971).Google Scholar
[14]Effros, E. G.. On multilinear completely bounded maps. Contemp. Math. 62 (1987), 450–479.Google Scholar
[15]Effros, E. G. and Kishimoto, A.. Module maps and Hochschild-Johnson cohomology. Indiana Math. J. 36 (1987), 257–276.Google Scholar
[16]Grothendieck, A.. Resume de la theorie metrique des produits tensoriels topologiques. Bull. Soc. Mat. Sao Paulo 8 (1956), 1–79.Google Scholar
[17]Haagerup, U.. Injectivity and decompositions of completely bounded maps. In Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. vol. 1132 (Springer-Verlag, 1983), pp. 170–222.Google Scholar
[18]Haagerup, U.. The Grothendieck inequality for bilinear forms on C*-algebras. Adv. in. Math. 56 (1985), No. 2, 93–116.Google Scholar
[19]Hadwin, D. W.. Dilations and Hahn decompositions for linear maps. Canad. J. Math. 33 (1981), 826–839.Google Scholar
[20]Itoh, T.. The maximal C*-norm and the Haagerup norm. Math. Proc. Cambridge Philos. Soc. 107 (1990), 109–114.Google Scholar
[21]Itoh, T.. The Haagerup type cross norm on C*-algebras. (Preprint.)Google Scholar
[22]Kaijser, S.. Some results in the metric theory of tensor products. U.U.D.M. Report No. 1973:2, Uppsala (1973).Google Scholar
[23]Kaijser, S. and Sinclair, A. M.. Projective tensor products of C*-algebras. Math. Scand. 65 (1984), 161–187.Google Scholar
[24]Kadison, R. V. and Ringrose, J. R.. Fundamentals of the Theory of Operator Algebras, vol. 1 (Academic Press, 1983).Google Scholar
[25]Paulsen, V. I.. Completely Bounded Maps and Dilations. Research Notes in Math. (Pitman, 1986).Google Scholar
[26]Paulsen, V. I. and Power, S. C.. Tensor products of non-self-adjoint operator algebras. Rocky Mountain J. Math. (To appear.)Google Scholar
[27]Paulsen, V. I. and Smith, R. R.. Multilinear maps and tensor norms on operator systems. J. Fund. Anal. 73 (1987), 258–276.Google Scholar
[28]Pisier, G.. Un theoreme sur les operateurs entre espaces de Banach qui se factorisent par un espace de Hilbert. Ann. École. Norm. Sup. 13 (1980), 23–43.Google Scholar
[29]Pister, G.. Factorization of Linear Operators and Geometry of Banach Spaces. CBMS series no. 60 (American Mathematical Society, 1986).Google Scholar
[30]Takesaki, M.. Theory of Operator Algebras, vol. 1 (Springer-Verlag, 1979).Google Scholar
[31]Tonge, A. M.. Sur les algebres de Banach et les operateurs p-sommants. Exposé no. 13, Seminaire Maurey-Schwarz, 1975–1976.Google Scholar
[32]Tonge, A. M.. Banach algebras and absolutely summing operators. Math. Proc. Cambridge Philos. Soc. 80 (1976), 465–473.Google Scholar
[33]Varopoulos, N. Th.. Tensor algebras and harmonic analysis. Acta Math. 119 (1967), 51–112.CrossRefGoogle Scholar
[34]Varopoulos, N. Th.. A theorem on operator algebras. Math. Scand. 37 (1975), 173–182.Google Scholar