Contractive perturbations in JB\(^*\)-triples (Q2880392)

This paper exploits in a fruitful way the connection between geometric properties and algebraic-topological ones in the setting of JB*-triples.NEWLINENEWLINELet \(S\) be a subset of the closed unit ball \(X_1\) of a Banach space \(X\). An element \(x\) in \(X_1\) is a contractive perturbation of \(S\) if \(| | x \pm s| | \leq 1\) for all \(s \in S\). The set \(\mathrm{cp}(S)\) of all contractive perturbations of \(S\) is a norm-closed convex subset of \(X_1\), and the mapping \(S \mapsto \mathrm{cp}(S)\) is a Galois connection on the subsets of \(X_1\), with associated closure operator \(\mathrm{cp}^{(2)}(S) :=\mathrm{cp}(\mathrm{cp}(S))\). The geometric rank of an element \(x \in X_1\) is defined as the dimension of the closed linear subspace generated by \(\mathrm{cp}(\{x\})\). Moreover, \(x \in X_1\) is said to be geometrically compact (geometrically weakly compact) if \(\mathrm{cp}(\{x\})\) is a compact (weakly compact) subset of \(X\), respectively.NEWLINENEWLINENEWLINE\textit{M. Anoussis} and \textit{E. G. Katsoulis} [Proc. Am. Math. Soc. 124, No. 7, 2115--2122 (1996; Zbl 0857.46034)] proved that an element \(a\) in the closed unit ball of a \(C^*\)-algebra \(A\) has finite geometric rank if and only it lies in the socle. Moreover, it can be derived from previous results that, in the setting of a \(C^*\)-algebra, the notions of geometrically compact, geometrically weakly compact, compact and weakly compact are equivalent, where \(a \in A\) is said to be compact (weakly compact) if the operator \(x \mapsto axa\), \(x \in A\), is compact (weakly compact, respectively).NEWLINENEWLINEIn the paper under review, the authors prove that an element \(x\) in the closed unit ball of a JB*-triple \(E\) is geometrically weakly compact if and only if it is weakly compact, i.e., the operator \(x \mapsto \{ a, x, a\}\), \(x \in E\), is weakly compact, and that, unlike for the \(^C*\)-case, there exist JB*-triples (for instance, any infinite-dimensional spin factor) whose closed unit balls contain compact elements which are not geometrically compact.