Additive bijections of \(C(X)\) (Q1807569)

The authors consider additive bijections \(H:C(X,K)\to C(Y,K)\), where \(X,Y\) are compact Hausdorff spaces, \(K=\mathbb R,\mathbb C\), or \(\mathbb Q_p\). It is assumed that \(H\) is separating, that is the equality \(fg=0\) for some \(f,g\in C(X,K)\) implies \((Hf)(Hg)=0\). It is proved that an additive separating bijection \(H\) is automatically continuous. If \(K=\mathbb R\) or \(\mathbb Q_p\), then \(Hf=(f\circ h)H(\mathbf 1)\), where \(\mathbf 1(x)\equiv 1\), and \(h:\;Y\to X\) is a homeomorphism. A similar (a little more complicated) representation is found also for \(K=\mathbb C\). An example shows that the above result does not hold for complete nontrivially valued non-Archimedean fields, other than \(\mathbb Q_p\).