Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the ranks of single elements of reflexive operator algebras

Author(s): W. E. Longstaff; Oreste Panaia
Journal: Proc. Amer. Math. Soc. 125 (1997), 2875-2882.
MSC (1991): Primary 47C05
MathSciNet review: 1402872
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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:

1.: T. Donnellan, Lattice theory, Pergamon Press, Oxford and New York, (1968). MR 38:2059
2.: M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, J. London Math. Soc. (2) 25 (1982), 499-512. MR 84f:47053
3.: M. S. Lambrou, Automatic continuity and implementation of homomorphisms, (manuscript).
4.: M. S. Lambrou, On the rank of operators in reflexive algebras, Linear Alg. & Applic. 142 (1990), 211-235. MR 91k:47104
5.: M. S. Lambrou and W. E. Longstaff, Spatiality of isomorphisms between certain reflexive algebras, Proc. Amer. Math. Soc. (4) 122 (1994), 1065-1073. MR 95b:47053
6.: W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (11) 2 (1975), 491-498. MR 52:15036
7.: G. N. Raney, Tight Galois connections and complete distributivity, Trans. Amer. Math. Soc. 97 (1960), 418-426. MR 22:10928
8.: J. R. Ringrose, On some algebras of operators II, Proc. London Math. Soc. (3) 16 (1966), 385-402. MR 33:4703

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