, Volume 71, Issue 4, pp 301-310

Diameter preserving linear bijections of C(X)

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Abstract.

The aim of this paper is to solve a linear preserver problem on the function algebra C(X). We show that in the case in which X is a first countable compact Hausdorff space, every linear bijection \(\phi :C(X)\to C(X)\) having the property that \(\hbox {diam} (\phi (f)(X))=\hbox {diam} (f(X)) (f\in C(X))\) is of the form¶¶ \( \phi (f)=\tau \cdot f\circ \varphi +t(f)1 \,\, (f\in C(X))\) ¶¶where \(\tau \in {\Bbb C}, |\tau |=1, \)\varphi :X\to X$ is a homeomorphism and \(t:C(X)\to {\Bbb C}\) is a linear functional.

Received: 29.8.1997