Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T01:32:22.312Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

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