Algebraic isomorphisms and -subspace lattices

Authors:
Jiankui Li and Oreste Panaia

Journal:
Proc. Amer. Math. Soc. **133** (2005), 2577-2587

MSC (2000):
Primary 47L10

Published electronically:
April 15, 2005

MathSciNet review:
2146201

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Abstract | References | Similar Articles | Additional Information

Abstract: The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -*subspace lattice*, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

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Additional Information

**Jiankui Li**

Affiliation:
Department of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
jli@math.uwaterloo.ca

**Oreste Panaia**

Affiliation:
School of Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia

Email:
oreste@maths.uwa.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-05-07581-7

Keywords:
Algebraic isomorphism,
rank-one operator,
single element

Received by editor(s):
February 4, 2002

Received by editor(s) in revised form:
April 17, 2003

Published electronically:
April 15, 2005

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society