On the ranks of single elements of reflexive operator algebras
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- by W. E. Longstaff and Oreste Panaia
- Proc. Amer. Math. Soc. 125 (1997), 2875-2882
- DOI: https://doi.org/10.1090/S0002-9939-97-03968-3
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Abstract:
For any completely distributive subspace lattice $\mathfrak {L}$ on a real or complex reflexive Banach space and a positive integer $n$, necessary and sufficient (lattice-theoretic) conditions are given for the existence of a single element of $Alg\mathfrak {L}$ of rank $n$. Similar conditions are given for the existence of single elements of infinite rank. From this follows a relatively simple lattice-theoretic condition which characterises when every non-zero single element has rank one. Slightly stronger results are obtained for the case where $\mathfrak {L}$ is finite, including the fact that every single element must then be of finite rank.References
- Thomas Donnellan, Lattice theory, Pergamon Press, Oxford-New York-Toronto, 1968. MR 0233738
- 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
- M. S. Lambrou, Automatic continuity and implementation of homomorphisms, (manuscript).
- M. S. Lambrou, On the rank of operators in reflexive algebras, Linear Algebra Appl. 142 (1990), 211–235. MR 1077986, DOI 10.1016/0024-3795(90)90268-H
- M. S. Lambrou and W. E. Longstaff, Spatiality of isomorphisms between certain reflexive algebras, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1065–1073. MR 1216818, DOI 10.1090/S0002-9939-1994-1216818-9
- W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), no. 4, 491–498. MR 394233, DOI 10.1112/jlms/s2-11.4.491
- George N. Raney, Tight Galois connections and complete distributivity, Trans. Amer. Math. Soc. 97 (1960), 418–426. MR 120171, DOI 10.1090/S0002-9947-1960-0120171-3
- J. R. Ringrose, On some algebras of operators. II, Proc. London Math. Soc. (3) 16 (1966), 385–402. MR 196516, DOI 10.1112/plms/s3-16.1.385
Bibliographic Information
- W. E. Longstaff
- Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia
- Email: longstaf@maths.uwa.edu.au
- Oreste Panaia
- Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia
- Email: oreste@maths.uwa.edu.au
- Received by editor(s): April 1, 1996
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2875-2882
- MSC (1991): Primary 47C05
- DOI: https://doi.org/10.1090/S0002-9939-97-03968-3
- MathSciNet review: 1402872