Diameter-preserving linear bijections of function spaces (Q2747264)

A mapping \(\Phi\) on some subspace \(A(X)\) of continuous functions on a compact Hausdorff space \(X\) into some other space of continuous functions on a compact Hausdorff space \(Y\) is said to be diameter-preserving if for every \(f\in A(X)\), \(\dim(f(X))=\dim(\Phi f(Y))\). When the paper under review was written, it was known that the diameter-preserving linear bijections on \(C(X)\) are all of the form \(\Phi(f)=\alpha f\circ\phi + \Lambda(f)\cdot 1\), where \(|\alpha|=1\), \(\phi\) is a homeomorphism on \(X\) and \(\Lambda\in C(X)^\ast\). The authors prove that this description is true for a wider class of situations by proving the following result: NEWLINENEWLINENEWLINETheorem. Let \(K_1\) and \(K_2\) be two compact convex sets such that every extreme point is a split face. Let \(A(K_i)\), \(i=1,2\), be closed subspaces of \(C(K_i)\) which contain the constants, separate points on \(K_i\) and which are self-adjoint. Then every diameter-preserving linear bijection \(\Phi:A(K_1)\rightarrow A(K_2)\) can be realized as \(\Phi(f)=\alpha f\circ\phi + \Lambda(f)\cdot 1\), where \(|\alpha|=1\), \(\phi:K_2 \rightarrow K_1\) is an affine homeomorphism and \(\Lambda\in A(K_1)^\ast\). NEWLINENEWLINENEWLINEAfter the main result has been proven, the authors also obtain such a description of diameter-preserving linear bijections on function algebras on their maximal ideal space. Next, they work out concrete formulas for the disk algebra, for the space of bounded continuous functions on a completely regular space and for the affine functions on the ``unit square'' in the plane. In this third example, the split face property does not hold, still the representation formula turns out to be valid. The third and last section of the paper is devoted to vector valued function spaces. Adjustments are made to obtain a result similar to the main theorem for diameter-preserving linear bijections from \(C(X,E)\) to \(C(X,F)\), where \(E\) and \(F\) are Banach spaces such that the one-dimensional span of any extreme point of the dual unit ball is an \(L\)-summand.