-algebras of topological stable rank 1 were introduced by M. A. Rieffel [Proc. Lond. Math. Soc. (3) 46, 301–333 (1983; Zbl 0533.46046)]. Rieffel also proved that a -algebra is of topological stable rank 1 (tsr 1) if and only if its invertible elements are norm dense. The paper under review considers this characterisation as starting point to extend the notion of topological stable rank 1 to the more general setting of -algebras.
The author proves, among other results, that a complex spin factor and a finite-dimensional -algebra have stable rank one, providing an example of a special -algebra of tsr 1 whose enveloping von Neumann algebra is not trs 1. The author also proves some interesting characterisations of invertible extreme points of the closed unit ball of a -algebra. Concretely, an extreme point of the closed unit ball of a unital -algebra is unitary if and only if its distance to the invertible elements is strictly less than 1. Consequently, given a -algebra of tsr 1 , the extreme points of the closed unit ball of and unitary elements in coincide. This characterisation extends an analogous results obtained by G. K. Pedersen in [J. Oper. Theory 26, No. 2, 345–381 (1991; Zbl 0784.46043)].