A new version of the nonarchimedean Banach-Stone theorem (Q2751721)

For two compact spaces \(X,Y\) and a complete non-archimedean (n.a) valued field \(\mathbb K\) denote by \(C^*(X), C^*(Y),\) the spaces of all \(\mathbb K\)-continuous bounded maps from \(X\), respectively \(Y\), with the sup norm \(\|\|_\infty \). A linear map \(T:C^*(X)\to C^*(Y)\) is called a Banach-Stone map provided there exists a homeomorphism \(h:Y\to X\) and \(a\in C^*(Y), |a|\equiv 1,\) such that \(Tf(y) = a(y)f(h(y)), \) for all \(y\in Y\) and all \(f\in C^*(X)\). The classical Banach-Stone theorem asserts that any isometric isomorphism \(T:C(X)\to C(Y)\) is a Banach-Stone map. The result is not longer true in n.a. case -- there exist non-homemorphic compact zero dimensional spaces \(X, Y\) such that \(C^*(X)\) and \(C^*(Y)\) are isometrically isomorphic [see \textit{A. C. M. van Rooij}, ``Non-Archimedean Functional Analysis'', New York (1978; Zbl 0396.46061)]. The aim of the present paper is to give necessary and sufficient conditions in order that a linear bijection between \(C^*(X) \) and \(C^*(X)\) be a Banach-Stone map. For \(f\in C^*(X)\) put \(v(f) = \{|f(x)|: x\in X\}\) and \(v^*(f) = v(f)\setminus \{0\}\). Then a linear bijective mapping \( T: C^*(X)\to C^*(Y)\) is a Banach-Stone map iff \(v^*(f) =\{1\} \) implies \(v^*(Tf) =\{1\} \), and iff \(\sup v(f) - \inf v(f) = \sup v(Tf) - \inf v(Tf)\), for all \(f\in C^*(X)\) (Th. 3). Also, if \(C^*(X), C^*(Y)\) are equipped with the \(A\)-norms (see loc. cit.) given by \(\|f\|= 2\|f\|_\infty - \inf v(f)\), then a bijective linear mapping \(T: C^*(X)\to C^*(Y)\) is Banach-Stone map iff it is an isometry with respect to the considered \(A\)-norms (Th. 4).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].