On 2-local nonlinear surjective isometries on normed spaces and \(C^*\)-algebras. (English) Zbl 1445.46012
Let \(X\) be a normed linear space. Let \(\tau{\mathcal L}(X)\) denote the set of self mappings \(\Phi\) (not necessarily linear) with the property that, for every \(a,b \in X\), there exists a surjective isometry \(\phi_{a,b}\) (not necessarily linear) such that \(\phi_{a,b}(a) = \Phi(a)\) and \(\phi_{a,b}(b)=\Phi(b)\). It turns out that such a \(\Phi\) need not be a surjection. It is so when \(X = C(K)\) for a first countable compact Hausdorff space \(K\). It is an affine map when the real span of the set of extreme points of the unit ball of \(X\) is dense in \(X\).
Reviewer: T.S.S.R.K. Rao (Bangalore)
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46L05 | General theory of \(C^*\)-algebras |
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