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Remarks on complemented subspaces of von Neumann algebras*

Published online by Cambridge University Press:  14 November 2011

G. Pisier
G. Pisier
Affiliation:
Mathematics Department, Texas A. and M. University, College Station, TX 77843, U.S.A.; and Equipe d'Analyse, Université Paris 6, Tour 46, 4ème étage, 75230 Paris Cedex 05, France
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Synopsis

In this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 1-2 , 1992 , pp. 1 - 4
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Arveson, W. B.. Subalgebras of C*-algebras. Acta Math. 123 (1969) 141–224.CrossRefGoogle Scholar
2Jarchow, H.. Weakly compact operators on C*-algebras. Math. Ann. 273 (1986) 341–343.CrossRefGoogle Scholar
3Pisier, G.. Factorization of operators through Lp∞ or Lpl and non-commutative generalizations. Math. Ann. 276 (1986) 105–136.CrossRefGoogle Scholar
4Takesaki, M.. Theory of Operator Algebras, vol. 1 (Berlin: Springer, 1979).CrossRefGoogle Scholar
5Tomiyama, J.. On the projection of norm one in W*-algebras. Proc. Japan Acad. 33 (1957) 608–612.Google Scholar