The Range of a Structural Projection

Abstract

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.

Exposed faces of the unit ball in a JBW*-triple

“…By [7], Lemma 2.1, B is an inner ideal in A. The converse is an immediate consequence of the same lemma.…”

“…Let v be a tripotent in J. By [7], Lemma 2.1, A 2 v is a subset of J. By hypothesis, the weak à -closed subtriple A 2 v A 0 u contains no nonzero tripotent and therefore coincides with f0g.…”

“…To prove (ii) let u be an element in u ' J and let M be an orthogonal subset of uA such that M u. Then, by [7], Lemma 2.1, M is a subset of J and, thus, M is at most countable. Therefore the tripotent u lies in u ' A J.…”

“…With respect to the separately weak à -continuous product aY b U 3 a b fa u bg and the norm-preserving involution a U 3 a y fu a ug, A 2 u is a JBW à -algebra with unit u. For details the reader is referred to [1], [3], [4], [5], [7], [10], [12], [13], [14], [15], [16], [17], [19], [20], [23].…”

Compact tripotents in bi-dual JB*-triples

“…The relationship between these three notions was studied in [11], [13] and [14] where the proof of the following result may be found. (…”