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Injective matricial Hilbert spaces

Published online by Cambridge University Press:  24 October 2008

A. Guyan Robertson
A. Guyan Robertson
Affiliation:
University of Edinburgh, Department of Mathematics, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ
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Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 183 - 190
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Arazy, J.. Linear topological classification of matroid C*-algebras. Math. Scand. 52 (1983), 89–111.CrossRefGoogle Scholar
[2]Berberian, S. K.. Baer *-rings (Springer-Verlag, 1972).CrossRefGoogle Scholar
[3]Bratteli, O. and Robinson, D.. Operator Algebras and Quantum Statistical Mechanics II (Springer-Verlag, 1981).CrossRefGoogle Scholar
[4]Choi, M. D. and Effros, E.. The completely positive lifting problem for C*-algebras. Ann. of Math. 104 (1976), 585–609.CrossRefGoogle Scholar
[5]Christbnsen, E. and Sinclair, A. M.. A survey of completely bounded operators. Bull. London Math. Soc. 21 (1989), 417–448.CrossRefGoogle Scholar
[6]Christensen, E. and Sinclair, A. M.. Completely bounded isomorphisms of injective von Neumann algebras. Proc. Edinburgh Math. Soc. 32 (1989), 317–327.CrossRefGoogle Scholar
[7]Effros, E. and Størmer, E.. Positive projections and Jordan structure in operator algebras. Math. Scand. 45 (1979), 127–138.CrossRefGoogle Scholar
[8]Fell, J. M. G.. The structure of algebras of operator fields. Acta Math. 106 (1961), 233–280.CrossRefGoogle Scholar
[9]Hanche-Olsen, H. and Størmer, E.. Jordan Operator Algebras. Pitman Monographs and Studies in Mathematics no. 21 (Pitman, 1984).Google Scholar
[10]Harris, L. A.. Bounded Symmetric Homogeneous Domains in Infinite Dimensional Spaces. Lecture Notes in Math. vol. 364 (Springer-Verlag, 1974), 13–40.Google Scholar
[11]Kaplansky, I.. Algebras of type I. Ann. of Math. 56 (1952), 460–472.CrossRefGoogle Scholar
[12]Loebl, R. I.. A Hahn decomposition for linear maps. Pacific J. Math. 65 (1976), 119–133.CrossRefGoogle Scholar
[13]Longo, R.. Simple injective subfactors. Adv. in Math. 63 (1987), 152–171.CrossRefGoogle Scholar
[14]Palmer, T. W.. Characterizations of C*-algebras. Trans. Amer. Math. Soc. 148 (1970), 577–588.Google Scholar
[15]Paulsen, V.. Completely Bounded Maps and Dilations. Pitman Research Notes in Mathematics no. 146 (Longman, 1986).Google Scholar
[16]Pedersen, G. K.. C*-algebras and Their Automorphism Groups (Academic Press, 1979).Google Scholar
[17]Roberts, J. E.. Cross products of von Neumann algebras by group duals. Sympos. Math. 20 (1976), 334–363.Google Scholar
[18]Robertson, A. G.. Strongly Chebyshev subspaces of matrices. J. Approx. Theory 55 (1988), 264–269.CrossRefGoogle Scholar
[19]Robertson, A. G. and Wassermann, S.. Completely bounded isomorphisms of injective operator systems. Bull. London Math. Soc. 21 (1989), 285–290.CrossRefGoogle Scholar
[20]Robertson, A. G. and Youngson, M. A.. Isomorphisms of injective operator spaces and Jordan triple systems. Quart. J. Math. Oxford Ser. (2) 41 (1990), 449–462.CrossRefGoogle Scholar
[21]Robertson, A. G. and Yost, D. T.. Chebyshev subspaces of operator algebras. J. London Math. Soc. 19 (1979), 523–531.CrossRefGoogle Scholar
[22]Ruan, Z. J.. Injectivity of operator spaces. Trans. Amer. Math. Soc. 315 (1989), 89–104.CrossRefGoogle Scholar
[23]Upmeier, H.. Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics. CBMS no. 67 (American Mathematical Society, 1987).CrossRefGoogle Scholar
[24]Blecher, D. P.. Tensor products of operator spaces II (preprint, 1990). University of Houston.Google Scholar
[25]Effros, E. and Ruan, Z.-J.. Self-duality for the Haagerup tensor product and Hilbert space factorizations (preprint, 1990).Google Scholar
[26]Jarchow, H.. Weakly compact operators on C*-algebras. Math. Ann. 273 (1986), 341–343.CrossRefGoogle Scholar
[27]Pisier, G.. Factorization of operators through Lp∞ or Lp1 and non-commutative generalizations. Math. Ann. 276 (1986), 105–136.CrossRefGoogle Scholar