Isometries of spaces of unbounded continuous functions (Q2730747)

Let \(X\) be a completely regular Hausdorff space and let \(C(X)\) denote the set of all continuous, scalar-valued functions defined on \(X\). The well-known Banach-Stone theorem establishes that, when \(X\) is compact, the Banach space structure of \(C(X)\) (equipped with the sup norm) completely characterizes the topological structure of \(X\). In this paper the authors deal with an extension of the Banach-Stone theorem to spaces of not necessarily bounded, continuous functions. Assume that \(X\) is a \(\sigma\)-compact space and let \(X_{1}\), \(X_{2}\),\dots be a fixed sequence of compact subsets such that \(X_{1}\varsubsetneq X_{2}\varsubsetneq \dots \varsubsetneq \dots \), and \(\cup_{n=1}^{\infty} X_{n} = X\). The two metrics considered here are: NEWLINE\[NEWLINE\text{(i)\;}\|f\|=\sum_{n=1}^{\infty} a_{n}\|f\|_{n}, \text{ and (ii)\;}d(f,g)=\sum_{n=1}^{\infty} a_{n}\frac{\|f-g\|_{n}}{1+\|f-g\|_{n}},NEWLINE\]NEWLINE where \(\|.\|_{n}\) is the usual sup norm on the set \(X_{n}\) and \((a_{n})_{n=1}^{\infty}\) is a fixed sequence of positive numbers in \(\ell^{1}\). The formula (i) defines a Banach space norm on a subspace of \(C(X)\) which contains unbounded functions of restricted rate of growth. This subspace is denoted by \(\overline{C}(X)\). The formula (ii) defines a complete metric on \(C(X)\). Using some interesting techniques, the following two results are obtained: NEWLINENEWLINENEWLINE(1) Let \((\overline{C}(X),\|.\|)\) be the Banach space defined by (i) on a \(\sigma\)-compact space \(X\) and let \((\overline{C}(X'),\|.\|)\) be the analogous on \(X'\). Assume that \(T\) is a linear isometry from \(\overline{C}(X)\) onto \(\overline{C}(X')\). Then there is a homeomorphism \(\varphi\) of \(X'\) onto \(X\) with \(\varphi(X'_{n})=X_{n}\), for all \(n\in \mathbb{N}\),and a continuous, scalar-valued function \(\kappa\) on \(X'\) such that \(Tf=\kappa f\circ \varphi\) for all \(f\in \overline{C}(X)\).NEWLINENEWLINENEWLINE(2) Let \((C(X),d(.,.))\) the metric vector space defined by (ii) and let \((C(X'),d(.,.))\) be the analogous on \(X'\). Assume that \(T\) is a linear isometry from \(C(X)\) onto \(C(X')\). Then there is a homeomorphism \(\varphi\) of \(X'\) onto \(X\) with \(\varphi(X'_{n})=X_{n}\), for all \(n\in \mathbb{N}\),and a continuous, scalar-valued function \(\kappa\) on \(X'\) such that \(Tf=\kappa f\circ \varphi\) for all \(f\in \overline{C}(X)\).