On the Geometry of the Unit Ball of a J B * -Triple

Abstract

We explore a $J{B}^{\mathrm{*}}$-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting of $C^{*}$-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 a $J{B}^{*}$-triple; this indicates their structural richness. We initiate a study of the unit ball of a $J{B}^{*}$-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. Some $C^{*}$-algebra and $J{B}^{*}$-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended to $J{B}^{*}$-triples.