A short proof of the result on actions that characterize \(\ell^n_\infty\)
The author gives a proof of a theorem due to \textit{B. Chalmers} and \textit{B. Shekhtman} [Linear Algebra Appl. 270, 155-169 (1998; Zbl 0898.46011)] respectively: Let \(T\) be an operator on an \(n\)-dimensional linear space \(L\) which is not a multiple of the identity. Then \(L\) can be endowed with a norm satisfying the following conditions: (1) The obtained \(n\)-dimensional normed space \(V\) is not isometric to \(\ell^n_\infty\); (2) For every isometric embedding of the space \(V\) into a Banach space \(X\) there exists a norm preserving extension \(\widetilde{T}\) of \(T\) onto \(X\). If the identity operator on an \(n\)-dimensional Banach space \(V\) can be extended to any Banach space with the same norm, then \(V\) is isometric to \(\ell^n_\infty\), and, the identity is the only such operator.
scientific article; zbMATH DE number 1371399