Automorphisms in spaces of continuous functions on Valdivia compacta

TL;DR

This paper investigates automorphisms in spaces of continuous functions on Valdivia compacta, showing that only c0(I) spaces are automorphic among certain classes, but subspace isomorphisms extend to automorphisms under specific conditions.

Contribution

It establishes the non-existence of automorphic C(K) spaces for Valdivia compacta except c0(I), and proves extension of subspace isomorphisms to automorphisms for certain Eberlein compacta.

Findings

No automorphic C(K) spaces for Valdivia compacta except c0(I)

Subspace isomorphisms extend to automorphisms in specific Eberlein compacta

C(K) not isomorphic to c0(I) implies certain extension properties

Abstract

We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(I). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(I), all isomorphism between subspaces of C(K) of size less than aleph_omega extend to automorphisms of C(K).