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$J{B}^{*}$-algebras of topological stable rank 1. (English) Zbl 1161.46041

${C}^{*}$-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 ${C}^{*}$-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 $J{B}^{*}$-algebras.

The author proves, among other results, that a complex spin factor and a finite-dimensional $J{B}^{*}$-algebra have stable rank one, providing an example of a special $JB{W}^{*}$-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 $J{B}^{*}$-algebra. Concretely, an extreme point of the closed unit ball of a unital $J{B}^{*}$-algebra is unitary if and only if its distance to the invertible elements is strictly less than 1. Consequently, given a $J{B}^{*}$-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 G. K. Pedersen in [J. Oper. Theory 26, No. 2, 345–381 (1991; Zbl 0784.46043)].

MSC:
 46L70 Nonassociative selfadjoint operator algebras