Weak limits of projections and compactness of subspace lattices

Wagner, Bruce H.

doi:10.1090/S0002-9947-1987-0911083-2

Weak limits of projections and compactness of subspace lattices
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by Bruce H. Wagner

Trans. Amer. Math. Soc. 304 (1987), 515-535

DOI: https://doi.org/10.1090/S0002-9947-1987-0911083-2

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Abstract:

A strongly closed lattice of projections on a Hilbert space is compact if the associated algebra of operators has a weakly dense subset of compact operators. If the lattice is commutative, there are necessary and sufficient conditions for compactness, one in terms of the structure of the lattice, and the other in terms of a measure on the lattice. There are many examples of compact lattices, and two main types of examples of noncompact lattices. Compactness is also related to the study of weak limits of certain projections.

References

An invitation to

algebras

William Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433–532. MR 365167, DOI 10.2307/1970956
William Arveson, Perturbation theory for groups and lattices, J. Funct. Anal. 53 (1983), no. 1, 22–73. MR 715546, DOI 10.1016/0022-1236(83)90045-9
William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
William Arveson, Subalgebras of $C^{\ast }$-algebras. II, Acta Math. 128 (1972), no. 3-4, 271–308. MR 394232, DOI 10.1007/BF02392166
Niels Toft Andersen, Compact perturbations of reflexive algebras, J. Functional Analysis 38 (1980), no. 3, 366–400. MR 593086, DOI 10.1016/0022-1236(80)90071-3
Kenneth R. Davidson, Commutative subspace lattices, Indiana Univ. Math. J. 27 (1978), no. 3, 479–490. MR 482264, DOI 10.1512/iumj.1978.27.27032
Kenneth R. Davidson, Similarity and compact perturbations of nest algebras, J. Reine Angew. Math. 348 (1984), 72–87. MR 733923, DOI 10.1515/crll.1984.348.72
Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition, revue et augmentée. MR 0352996
Thomas Fall, William Arveson, and Paul Muhly, Perturbations of nest algebras, J. Operator Theory 1 (1979), no. 1, 137–150. MR 526295
C. Foiaş, C. Pasnicu, and D. Voiculescu, Weak limits of almost invariant projections, J. Operator Theory 2 (1979), no. 1, 79–93. MR 553865
John Froelich, Compact operators in the algebra of a partially ordered measure space, J. Operator Theory 10 (1983), no. 2, 353–355. MR 728914

Compact operators, invariant subspaces, and spectral synthesis

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
A. Guichardet, Produits tensoriels infinis et représentations des relations d’anticommutation, Ann. Sci. École Norm. Sup. (3) 83 (1966), 1–52 (French). MR 0205097
Alan Hopenwasser, Cecelia Laurie, and Robert Moore, Reflexive algebras with completely distributive subspace lattices, J. Operator Theory 11 (1984), no. 1, 91–108. MR 739795
M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, J. London Math. Soc. (2) 25 (1982), no. 3, 499–512. MR 657507, DOI 10.1112/jlms/s2-25.3.499
David R. Larson, Nest algebras and similarity transformations, Ann. of Math. (2) 121 (1985), no. 3, 409–427. MR 794368, DOI 10.2307/1971180
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
Cecelia Laurie, Complete distributivity and ordered group lattices, Proc. Amer. Math. Soc. 95 (1985), no. 1, 79–82. MR 796450, DOI 10.1090/S0002-9939-1985-0796450-6
Cecelia Laurie and W. E. Longstaff, A note on rank-one operators in reflexive algebras, Proc. Amer. Math. Soc. 89 (1983), no. 2, 293–297. MR 712641, DOI 10.1090/S0002-9939-1983-0712641-2
Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. MR 0355639
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Donald Sarason, On spectral sets having connected complement, Acta Sci. Math. (Szeged) 26 (1965), 289–299. MR 188797
Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338

On infinite direct products

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