Structural Projections on JBW*-Triples

Abstract

A linear projection R on a Jordan*-triple A is said to be structural provided that, for all elements a, b and c in A, the equality {Rab Re} = R{a Rbc} holds. A subtriple B of A is said to be complemented if A = B + Ker (B), where Ker(B) = {aeA: {BaB} = 0}. It is shown that a subtriple of a JBW*-triple is complemented if and only if it is the range of a structural projection.A weak* closed subspace B of the dual E* of a Banach space E is said to be an N*-ideal if every weak* continuous linear functional on B has a norm preserving extension to a weak* continuous linear functional on E* and the set of elements in E which attain their norm on the unit ball in B is a subspace of E. It is shown that a subtriple of a JBW*-triple A is complemented if and only if it is an N*-ideal, from which it follows that complemented subtriples of A are weak* closed, and structural projections on A are weak* continuous and norm non-increasing. It is also shown that every N*-ideal in A possesses a triple product with respect to which it is a JBW*-triple which is isomorphic to a complemented subtriple of A.

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

“…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. (…”

“…When A s exhausts A or, equivalently, when the open unit ball in A is a bounded symmetric domain, the complex Banach space A is said to be a JB*-triple, the properties of which have received much attention in recent years. See, for example, [2], [6][7][8][9][10][11][12][13], [15][16][17], [19], [30], [31].…”

Generalized n-circular projections on JB*-triples and Hilbert C0(Ω)-modules

“…We shall say that a subtriple J 1 is complementary to J 2 if ker. [20] for details. We also refer the readers to [17] for the general theory of JB*-triples.…”

“…Ever since the papers [20] and [42], various classes of projections on JB*-triples attract many attention in literature. According to [22,Theorem 4], see also [29, Theorem 2.1], every generalized bicircular projection on a JB*-triple is generalized orthogonal or hermitian (hence also generalized orthogonal).…”