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The Range of a Structural Projection

Authors
Journal
Journal of Functional Analysis
0022-1236
Publisher
Elsevier
Publication Date
Volume
139
Issue
1
Identifiers
DOI: 10.1006/jfan.1996.0083
Disciplines
  • Mathematics

Abstract

Abstract Let Abe a JBW*-triple. A linear subspace Jof Ais called an inner idealin Aprovided that the subspace { J A J} is contained in J. A subtriple Bin Ais said to be complementedif A= B⊕Ker( B), where Ker( B)={ a∈ A : { B a B}=0}. A complemented subtriple in Ais a weak*-closed inner ideal. A linear projection on Ais said to be structuralif, for all elements a, band cin A,[formula]The range of a structural projection is a complemented subtriple and, conversely, a complemented subtriple is the range of a unique structural projection. We analyze the structure of the weak*-closed inner ideal generated by two arbitrary tripotents in a JBW*-triple in terms of the simultaneous Peirce spaces of three suitably chosen pairwise compatible tripotents. This result is then used to show that every weak* closed inner ideal Jin a JBW*-triple Ais a complemented subtriple in Aand therefore the range of a unique structural projection on A. As an application structural projections on W*-algebras are considered.

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