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