Collinear Systems and Normal Contractive Projections on JBW*-Triples

Abstract

Given a family {x k } k∈K of elements x k in the predual A * of a JBW * -triple A, such that the support tripotents e k of x k form a collinear system in the sense of [31], necessary and sufficient criteria for the existence of a contractive projection from A * onto the subspace lin{x k : k ∈ K} n are provided. Preparatory to these results, and interesting in itself, is a set of necessary and sufficient algebraic conditions upon a contractive projection P on A for its range P A to be a subtriple. The results also provide criteria for the range of a normal contractive projection on A to be a Hilbert space.

Transitivity of inner automorphisms in infinite dimensional Cartan factors

“…The partial isometries of B(H, K) are precisely the tripotent elements. In [16] Loos showed that when C is a finite-dimensional Cartan factor, then the group Inn(C) acts transitively on the set U r (C) of tripotents of rank r in C. An analogous statement for Hilbert spaces of arbitrary dimension was proved in [11]. The goal of the present article is to generalize that result to all Cartan-factors.…”

“…The above-cited works show that the geodesics in those manifolds are given by paths of inner automorphisms of the corresponding JBW * -triple. The arguments presented in [11] and [12] show that the transitivity of Inn(C) on U 1 (C) is closely connected to the problem of the existence of contractive projections onto subtriples of JB * -triples or JBW * -triples.…”

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

“…Here we consider atomic subtriples with finite-dimensional Cartan-factors of any of the six types in a general JBW * -triple A. We extend those main results of [21] concerning the relationships of contractive projections on A and on the pre-dual A * with the group Inn(A) of inner automorphisms and its Banach-Lie algebra inn(A) of inner derivations of A.…”

“…This underpins the principle of commutation as a simple algebraic characterization of the geometric property of contractivity. The said principle was first observed in the context of Hilbert spaces equipped with the Jordan-triple product of type-I Cartan-factors [21]. Here we consider atomic subtriples with finite-dimensional Cartan-factors of any of the six types in a general JBW * -triple A.…”

Characterizations of tripotents in JB*-triples

“…Part of the main results presented in this article were obtained in the authors Ph.D. thesis [15]. He wishes to acknowledge the advise of the late G. T. Rüttimann, University of Berne, and of C. M. Edwards, University of Oxford, as well as the support received by H. Carnal, University of Berne.…”