Nonlinear Diameter Preserving Maps Between Certain Function Spaces

Abstract

Let X, Y be compact Hausdorff spaces and A, B be subspaces of C(X) and C(Y), respectively, containing the constant functions such that B is point separating and the evaluation functionals are linearly independent on B. In this paper, we give the general form of a surjective, not assumed to be linear, diameter preserving map ${T:A \longrightarrow B}$ for the case where A is dense in C(X). Fixing a point ${x_1\in X}$, we show that there exist a subset ${Y_0}$ of Y, a scalar ${\beta\in \mathbb{T}}$, a bijective continuous map ${\Psi: Y_0 \longrightarrow X}$ and a constant function ${\alpha: Y_0 \longrightarrow \{-1,1\}}$ such that

$$\begin{aligned}T_{1} f(y) - T_{1} f(y_{1}) = & \beta ({\rm Re} (f(\Psi(y)) - f(\Psi(y_{1}))) \\ & + \alpha(y) i {\rm Im} (f(\Psi(y)) - f(\Psi(y_{1}))))\end{aligned}$$

for all ${f\in A}$ and ${y\in Y_0}$, where ${T_1=T-T0}$ and ${\Psi(y_1)=x_1}$. In particular, either

$$T_1(f)(y)=\beta f(\Psi(y))+L(f) \qquad (f\in A,y\in Y_0),$$

or

$$T_1(f)(y)=\beta \overline{f(\Psi(y))}+L(f) \qquad (f\in A, y\in Y_0),$$

holds for some functional L on A, which is linear (resp. real-linear) whenever T is so.