A note on compact ideal perturbations in semifinite von Neumann algebras
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- by Florin Pop PDF
- Proc. Amer. Math. Soc. 119 (1993), 843-847 Request permission
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
- Hari Bercovici and Florin Pop, On perturbations of reflexive algebras, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 379–387. MR 1065836, DOI 10.1216/rmjm/1181073113
- Kenneth R. Davidson and Stephen C. Power, Best approximation in $C^\ast$-algebras, J. Reine Angew. Math. 368 (1986), 43–62. MR 850614
- Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
- Thomas Fall, William Arveson, and Paul Muhly, Perturbations of nest algebras, J. Operator Theory 1 (1979), no. 1, 137–150. MR 526295 J. Froelich, Compact operators, invariant subspaces and spectral synthesis, Ph.D. Thesis, University of Iowa, 1984.
- Frank Gilfeather, Alan Hopenwasser, and David R. Larson, Reflexive algebras with finite width lattices: tensor products, cohomology, compact perturbations, J. Funct. Anal. 55 (1984), no. 2, 176–199. MR 733915, DOI 10.1016/0022-1236(84)90009-0
- Frank Gilfeather and David R. Larson, Nest-subalgebras of von Neumann algebras, Adv. in Math. 46 (1982), no. 2, 171–199. MR 679907, DOI 10.1016/0001-8708(82)90022-6
- V. Kaftal, D. Larson, and G. Weiss, Quasitriangular subalgebras of semifinite von Neumann algebras are closed, J. Funct. Anal. 107 (1992), no. 2, 387–401. MR 1172032, DOI 10.1016/0022-1236(92)90115-Y
- Cecelia Laurie, On density of compact operators in reflexive algebras, Indiana Univ. Math. J. 30 (1981), no. 1, 1–16. MR 600028, DOI 10.1512/iumj.1981.30.30001
- Sorin Popa and Florin Rădulescu, Derivations of von Neumann algebras into the compact ideal space of a semifinite algebra, Duke Math. J. 57 (1988), no. 2, 485–518. MR 962517, DOI 10.1215/S0012-7094-88-05722-5
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 119 (1993), 843-847
- MSC: Primary 46L10
- DOI: https://doi.org/10.1090/S0002-9939-1993-1184084-8
- MathSciNet review: 1184084