On the ranks of single elements

of reflexive operator algebras

Authors:
W. E. Longstaff and Oreste Panaia

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

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Abstract | References | Similar Articles | Additional Information

Abstract: For any completely distributive subspace lattice on a real or complex reflexive Banach space and a positive integer , necessary and sufficient (lattice-theoretic) conditions are given for the existence of a single element of of rank . 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 is finite, including the fact that every single element must then be of finite rank.

**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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Additional 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

DOI:
https://doi.org/10.1090/S0002-9939-97-03968-3

Received by editor(s):
April 1, 1996

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society