International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 37186, 24 pages
-Algebras of Topological Stable Rank 1
Department of Mathematics, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
Received 13 October 2005; Revised 17 December 2006; Accepted 5 February 2007
Academic Editor: Lokenath Debnath
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.
In 1976, Kaplansky introduced the class -algebras which includes all -algebras as a proper subclass. The notion of topological stable rank 1 for -algebras was originally introduced by M. A. Rieffel and was extensively
studied by various authors. In this paper, we extend this notion to general
-algebras. We show that the complex spin factors are of tsr 1 providing an example of special -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 -algebra is positive in certain isotope of ; if the algebra is finite-dimensional, then it is of tsr 1 and every element of is positive in some unitary isotope of . Further, it is established that extreme points of the unit ball
sufficiently close to invertible elements in a -algebra must be unitaries and that in any -algebras of tsr 1, all extreme points of the unit ball are unitaries. In the end, we prove the coincidence between the
-function and -function on invertibles in a -algebra.