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$JB^{\ast}$-algebras of topological stable rank 1. (English) Zbl 1161.46041
$C^*$-algebras of topological stable rank 1 were introduced by {\it 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 {\it G. K. Pedersen} in [J. Oper. Theory 26, No. 2, 345--381 (1991; Zbl 0784.46043)].

MSC:
46L70Nonassociative selfadjoint operator algebras
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Full Text: DOI EuDML
References:
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