We establish new estimates to compute the λ-function of Aron and Lohman on the unit ball of a JB * -triple. It is established that for every Brown-Pedersen quasi-invertible element a in a JB * -triple, for every Brown-Pedersen quasi-invertible element a in E1, where mq(a) is the square root of the quadratic conorm of a. For an element a in E1 which is not Brown-Pedersen quasi-invertible we can only estimate that λ(a) ≤ 1 2 (1−αq(a)). A complete description of the λ-function on the closed unit ball of every JBW * -triple is also provided, and as a consequence, we prove that every JBW * -triple satisfies the uniform λ-property.
We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting ofC*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in aJB*-triple; this indicates their structural richness. We initiate a study of the unit ball of aJB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. SomeC*-algebra andJB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended toJB*-triples.
Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.A short exact sequence is a diagram 0 → Y i − → X q − → Z → 0 of quasi Banach spaces and bounded linear operators such that the kernel of each arrow coincides with the image of the preceding one. The open mapping theorem implies that X contains i(Y ) and the quotient X/i(Y ) is isomorphic to Z. We shall also say that X is a twisted sum of Y and Z (or that X is an extension of Y by Z). The twisted sum X is said to be trivial if i(Y ) is complemented in X; otherwise, X is nontrivial.
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