## Abstract and Applied Analysis

### On the Geometry of the Unit Ball of a $J{B}^{\mathrm{*}}$-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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 891249, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511914

Digital Object Identifier
doi:10.1155/2013/891249

Mathematical Reviews number (MathSciNet)
MR3064400

#### Citation

Tahlawi, Haifa M.; Siddiqui, Akhlaq A.; Jamjoom, Fatmah B. On the Geometry of the Unit Ball of a $J{B}^{\mathrm{*}}$ -Triple. Abstr. Appl. Anal. 2013 (2013), Article ID 891249, 8 pages. doi:10.1155/2013/891249. https://projecteuclid.org/euclid.aaa/1393511914

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