Diameter preserving linear maps and isometries
Let \({\mathbb K}\) be \({\mathbb R}\) or \({\mathbb C}\), \(C(X)\) be the space of \({\mathbb K}\)-valued continuous functions on the compact Hausdorff space and \({C_0}(X)\) be the space of \({\mathbb K}\)-valued continuous functions on the locally compact noncompact space \(X\) vanishing at infinity. It is proved that a) a linear bijection \(T: C(X)\to C(X)\) is diameter preserving if and only if there is a homeomorphism \(\varphi : X\to X\), a linear functional \(\mu : C(X) \to \) \({\mathbb K}\) and a number \(\tau \) with \(|\tau |= 1\) and \(\mu (1_X) + \tau \not= 0\) such that \(Tf = \tau f\circ \varphi + \mu (f)1_X\) for every \(f\in C(X)\) (in [1] and [2] in References this result has been proved in the case of first countable compact Hausdorff space \(X\)); b) a linear bijection \(T : {C_0}(X)\to {C_0}(X)\) is diameter preserving if and only if there is a homeomorphism \(\varphi \) of \(X\cup \infty \) and a number \(\tau \) with \(|\tau |= 1\) such that \(Tf = \tau (f\circ \varphi - f(\varphi (\infty))1_{X\cup \infty })\) for each \(f\in {C_0}(X)\); c) the following statements are equivalent: (1) \(X\) and \(Y\) are homeomorphic; (2) \(C(X)\) and \(C(Y)\) are isomorphic; (3) \(C(X)/\ker \rho \) and \(C(Y)/\ker \rho \) are isometric (here \(\rho(f) = \sup \{|f(x) - f(y)|: x,y\in X\}\) for each \(f\in C(X)\)) and (4) there is a (not necessarily continuous) diameter preserving linear bijection \(C(X)\to C(Y)\). By an example is shown that the existence of a diameter preserving bijection between \({C_0}(X)\) and \({C_0}(Y)\) does not imply that \(X\) and \(Y\) are homeomorphic.
scientific article; zbMATH DE number 1383745