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Compact tripotents in bi-dual JB*-triples1

Published online by Cambridge University Press:  24 October 2008

C. Martin Edwards  and
Gottfried T. Rüttimann
C. Martin Edwards
Affiliation:
The Queen's College, Oxford
Gottfried T. Rüttimann
Affiliation:
University of Berne, Berne, Switzerland
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Abstract

The set consisting of the partially ordered set of tripotents in a JBW*-triple C with a greatest element adjoined forms a complete lattice. This paper is mainly concerned with the situation in which C is the second dual A** of a complex Banach space A and, more particularly, when A is itself a JB*-triple. A subset of consisting of the set of tripotents compact relative to A (denned in Section 4) with a greatest element adjoined is studied. It is shown to be an atomic complete lattice with the properties that the infimum of an arbitrary family of elements of is the same whether taken in or in and that every decreasing net of non-zero elements of has a non-zero infimum. The relationship between the complete lattice and the complete lattice where B is a Banach space such that B** is a weak*-closed subtriple of A** is also investigated. When applied to the special case in which A is a C*-algebra the results provide information about the set of compact partial isometries relative to A and are closely related to those recently obtained by Akemann and Pedersen. In particular it is shown that a partial isometry is compact relative to A if and only if, in their terminology, it belongs locally to A. The main results are applied to this and other examples.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 155 - 173
Copyright
Copyright © Cambridge Philosophical Society 1996

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