\(JB^{\ast}\)-algebras of topological stable rank 1
\(C^*\)-algebras of topological stable rank 1 were introduced by \textit{M. A. Rieffel} [Proc. Lond. Math. Soc. (3) 46, 301--333 (1983; Zbl 0533.46046)]. Rieffel also proved that a \(C^*\)-algebra \(A\) 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 \(JB^*\)-algebras. The author proves, among other results, that a complex spin factor and a finite-dimensional \(JB^*\)-algebra have stable rank one, providing an example of a special \(JBW^*\)-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 \(JB^*\)-algebra. Concretely, an extreme point of the closed unit ball of a unital \(JB^*\)-algebra is unitary if and only if its distance to the invertible elements is strictly less than 1. Consequently, given a \(JB^*\)-algebra of tsr 1 \(A\), the extreme points of the closed unit ball of \(A\) and unitary elements in \(A\) coincide. This characterisation extends an analogous results obtained by \textit{G. K. Pedersen} in [J.~Oper. Theory 26, No. 2, 345--381 (1991; Zbl 0784.46043)].
scientific article; zbMATH DE number 5282473
scientific article