Automorphisms in spaces of continuous functions on Valdivia compacta (Q952605)

A Banach space \(X\) is said to be automorphic if, for every isomorphism \(T:Y_1\to Y_2\) between two (closed) subspaces of \(X\) with \(\text{dens}(X/Y_1)=\text{dens}(X/Y_2)\), there exists an automorphism \(\widetilde{T}:X\to X\) which extends \(T\). The authors consider the well-known open question whether the only automorphic Banach spaces are \(\ell_2(\Gamma)\) and \(c_0(\Gamma)\). The main result of the paper states that there are no automorphic Banach spaces of the form \(C(K)\) with \(K\)-continuous image of a Valdivia compact except \(c_0(\Gamma)\). Nevertheless, there exists an Eberlein compact \(K\) of finite height such that \(C(K)\) is not isomorphic to \(c_0(\Gamma)\) and all isomorphisms between subspaces of \(C(K)\) of size less than \(\aleph_{\omega}\) extend to automorphisms of \(C(K)\).