Akhlaq A. Siddiqui

Copyright © 2007 Akhlaq A. Siddiqui. 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

In 1976, Kaplansky introduced the class JB*-algebras which includes all C*-algebras as a proper subclass. The notion of topological stable rank 1 for C*-algebras was originally introduced by M. A. Rieffel and was extensively studied by various authors. In this paper, we extend this notion to general JB*-algebras. We show that the complex spin factors are of tsr 1 providing an example of special JBW*-algebras for which the enveloping von Neumann algebras may not be of tsr 1. In the sequel, we prove that every invertible element of a JB*-algebra 𝒥 is positive in certain isotope of 𝒥; if the algebra is finite-dimensional, then it is of tsr 1 and every element of 𝒥 is positive in some unitary isotope of 𝒥. Further, it is established that extreme points of the unit ball sufficiently close to invertible elements in a JB*-algebra must be unitaries and that in any JB*-algebras of tsr 1, all extreme points of the unit ball are unitaries. In the end, we prove the coincidence between the λ-function and λu-function on invertibles in a JB*-algebra.