The Range of a Structural Projection

✍ Scribed by C.M. Edwards; K. McCrimmon; G.T. Rüttimann

• Book ID: 102972297
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 756 KB
• Volume: 139
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0083

✦ Synopsis

Let A be a JBW*-triple. A linear subspace J of A is called an inner ideal in A provided that the subspace

in A is a weak*-closed inner ideal. A linear projection on A is said to be structural if, for all elements a, b and c in A, [P a b Pc]=P[a Pb c].

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 J in a JBW*-triple A is a complemented subtriple in A and therefore the range of a unique structural projection on A. As an application structural projections on W*-algebras are considered.