A counterexample to Wood's conjecture
Let \(X\) be a Banach space and let \({\mathcal G}(X)\) denote the group of all surjective isometries of \(X\). \(X\) is said to be almost transitive if for every \(x_1,x_2 \in X\) with \(\| x_1\| = \| x_2\| =1 \) and \(\epsilon > 0\), there exists a \(T \in {\mathcal G}(X)\) such that \(\| T(x_1) - x_2 \| < \epsilon\). Let \(L\) be a locally compact Hausdorff space with more than one point and let \(C_0(L)\) denote the space of continuous functions vanishing at \(\infty\). \({\mathcal G}(C_0(L))\) is described by the classical Banach--Stone theorem. \textit{G.~V.\ Wood} [Proc.\ R.\ Ir.\ Acad., Sect.\ A 82, 177--186 (1982; Zbl 0495.46014)] proved that if \(C_0(L)\) is almost transitive, then \(L\) cannot be compact and its one point compactification must be connected. He conjectured that \(C_0(L)\) is never almost transitive. An affirmative answer when the scalar field is real was given by \textit{P.~Greim} and \textit{M.~Rajagopalan} [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 121, 75--80 (1997; Zbl 0901.46020)]. The present paper settles Wood's conjecture in the negative in the case of the complex scalar field by showing that if the one point compactification of \(L\) is a pseudoarc, then \(C_0(L)\) is almost transitive. The paper also shows how to construct such an \(L\) using the theory of continua. It is noted that a similar example also appears in the work of \textit{K.~Kawamura} [Glasg.\ Math.\ J.\ 47, 1--5 (2005; Zbl 1065.54006)]. The construction from this paper and the technique of ultrapowers have recently been used by \textit{F.~Cabello Sánchez} [Math.\ Z.\ 251, 735--749 (2005; Zbl 1089.46018)] to give an example of a locally compact set \(L\) and a `local isometry' \(\Phi: C_0(L) \rightarrow C_0(L)\), \(f,g \in C_0(L)\) with \(fg = 0\) but \(\Phi(f)\Phi(g) \neq 0\). This implies, in particular, that \(\Phi^\ast\) need not map the extreme points of the dual unit ball to extreme points, thus answering a question raised by the reviewer and \textit{K.~Jarosz} [Math.\ Z.\ 243, 449--469 (2003; Zbl 1027.46008)].
scientific article; zbMATH DE number 5017407