About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 891249, 8 pages
http://dx.doi.org/10.1155/2013/891249
Research Article

On the Geometry of the Unit Ball of a -Triple

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 14 February 2013; Revised 5 May 2013; Accepted 8 May 2013

Academic Editor: Ngai-Ching Wong

Copyright © 2013 Haifa M. Tahlawi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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