J B * -Algebras of Topological Stable Rank 1

In 1976, Kaplansky introduced the class J B * -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 J B * -algebras. We show that the complex spin factors are of tsr 1 providing an example of special J B W * -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 J B * -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 J B * -algebra must be unitaries and that in any J B * -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 λ u -function on invertibles in a J B * -algebra.