Diameter preserving linear maps and isometries. II (Q1570006)

Let \({\mathcal A}(S)\) be the space of all (real or complex) affine continuous functions on a compact convex set \(S\) in a locally convex Hausdorff space. A linear bijection \(T\) of \({\mathcal A}(S)\) is called diameter preserving if \(\rho (Tf) =\rho (f)\), where, for \(g\in {\mathcal A}(S)\), \(\rho (g) = \sup \{|g(x)-g(y)|: x,y, \in S\}\) denotes the diameter of the range of \(g\). The main result of the paper asserts that if \(S\) is a simplex then a linear bijection \(T\) of \({\mathcal A}(S)\) is diameter preserving iff there are an affine automorphism \(\phi :S\to S\), a linear functional \(\mu :{\mathcal A}(S)\to \mathbb K\), and a number \(\tau\) with \(|\tau|=1\) and \(\mu(1_S) + \tau \neq 0,\) such that \(Tf =\tau f\circ \phi +\mu (f)1_S, \) for \(f\in {\mathcal A}(S)\). Similar results for the space \(C(X)\) have been obtained by \textit{M. Györy} and \textit{L. Molnár} [Arch. Math. 71, No.~4, 301-310 (1998; Zbl 0928.46034)], and by the author in the first part of the present paper [Arch. Math. (Basel) 73, No. 5, 373-379 (1999)].