A characterization of locally compact spaces with homeomorphic one-point compactifications
Let \(X\) and \(Y\) be locally compact noncompact spaces. It is first shown that the set of extreme points of the closed unit ball of \(C_0(X)*\) consists exactly of \(\{\epsilon_x, - \epsilon_x, \epsilon_x - \epsilon_y : x,y \in X, x \neq y\}\), where \(\epsilon_x\) denotes the evaluation functional at \(x\). Using this result it is shown that there exist basically two types of diameter preserving linear bijections defined between a Borel continuous subspace \(A\) of \(C_0(X)\) and such a subspace \(B\) of \(C_0(Y).\) Type 1 consists of those which can be written as \(\psi(f)(y) = \tau(f \circ \phi)(y),\) where \(\tau \in \{-1, 1\},\) and \(\phi : Y \to X\) is a surjective homeomorphism. Type (2) consists of those which can be represented as \[ \psi(f)(y) = \left\{ \begin{matrix} \tau(f \circ \phi)(y) - \tau f(x_0), & {y \neq y_0},\\ - \tau f(x_0), & {y = y_0}, \end{matrix} \right. \] for some \(x_0 \in X\), \(y_0 \in Y\), where \(\phi : Y \backslash \{y_0\} \to X \backslash \{x_0\} \) is a homeomorphism which can be extended to the one-point compactifications of \(Y\) and \(X\). Finally, a characterization is given of locally compact spaces with homeomorphic one-point compactifications as those which admit diameter-preserving linear bijections like the ones described above.
scientific article; zbMATH DE number 1787518