Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on compact ideal perturbations in semifinite von Neumann algebras

Author: Florin Pop
Journal: Proc. Amer. Math. Soc. 119 (1993), 843-847
MSC: Primary 46L10
MathSciNet review: 1184084
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a semifinite von Neumann algebra and denote by $ J(M)$ the closed two-sided ideal generated by the finite projections in $ M$. A subspace $ S \subset M$ is called local if it is equal to the ultraweak closure of $ S \cap J(M)$. If $ M = B(H)$ and $ J(M) = K(H)$, Fall, Arveson, and Muhly proved that $ S + J(M)$ is closed for every local subspace $ S$.

In this note we prove that if $ M$ is a type $ {\text{I}}{{\text{I}}_\infty }$, factor, then there exist local subspaces in $ M$ which fail to have closed compact ideal perturbations; thus answering in the negative a question of Kaftal, Larson, and Weiss.

References [Enhancements On Off] (What's this?)

  • [1] H. Bercovici and F. Pop, On perturbations of reflexive algebras, Rocky Mountain J. Math. 20 (1990), 379-387. MR 1065836 (91k:47102)
  • [2] K. R. Davidson and S. C. Power, Best approximation in $ {C^{\ast}}$-algebras, J. Reine Angew. Math. 368 (1986), 43-62. MR 850614 (87k:47100)
  • [3] J. Dixmier, Les $ {C^{\ast}}$-algèbres et leur représentations, Gauthier-Villars, Paris, 1964. MR 0171173 (30:1404)
  • [4] T. Fall, W. Arveson, and P. Muhly, Perturbations of nest algebras, J. Operator Theory 1 (1979), 137-150. MR 526295 (80f:47035)
  • [5] J. Froelich, Compact operators, invariant subspaces and spectral synthesis, Ph.D. Thesis, University of Iowa, 1984.
  • [6] F. Gilfeather, A. Hopenwasser, and D. R. Larson, Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations, J. Funct. Anal. 55 (1984), 176-199. MR 733915 (85g:47062)
  • [7] F. Gilfeather and D. R. Larson, Nest-subalgebras of von Neumann algebras, Adv. in Math. 46 (1982), 171-199. MR 679907 (84c:47047)
  • [8] V. Kaftal, D. R. Larson, and G. Weiss, Quasitriangular subalgebras of semifinite von Neumann algebras are closed, J. Funct. Anal. 107 (1992), 387-401. MR 1172032 (93g:46060)
  • [9] C. Laurie, On density of compact operators in reflexive algebras, Indiana Univ. Math. J. 30 (1981), 1-16. MR 600028 (82b:47058)
  • [10] S. Popa and F. Radulescu, Derivations of von Neumann algebras into the compact ideal space of a semifinite algebra, Duke Math. J. 57 (1988), 485-518. MR 962517 (90a:46165)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L10

Retrieve articles in all journals with MSC: 46L10

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society