arXiv:0903.0658v1 [math.FA] 3 Mar 2009 AUTOMORPHISMS IN SPACES OF CONTINUOUS FUNCTIONS ON VALDIVIA COMPACTA ANTONIO AVIL ´ ES AND YOLANDA MORENO 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 c 0 (Γ). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c 0 (Γ), all isomorphism between subspaces of C(K) of size less than ℵω extend to automorphisms of C(K). Introduction A Banach space X is said to be automorphic if for every isomorphism T : Y 1 -→ Y 2 between two (closed) subspaces of X with dens(X/Y 1 )= dens(X/Y 2 ) there ex- ists an automorphism ˜ T : X -→ X which extends T , that is, ˜ T | Y1 = T . It has been shown in [9] that a necessary condition for a Banach space X to be automorphic is to be extensible, which means that for every subspace E ⊂ X and every operator T : E -→ X , there exists an operator ˜ T : X -→ X that extends T . Clearly every Hilbert space ℓ 2 (Γ) is automorphic and on the other hand, Lindenstrauss and Rosenthal [7] have proven that c 0 is automorphic and also that ℓ ∞ has a partial automorphic character, namely that isomorphisms T : Y 1 -→ Y 2 can be extended provided that ℓ ∞ /Y i is nonreflexive for i =1, 2, though ℓ ∞ is not automorphic. Moreno and Plichko [9] have recently shown that c 0 (Γ) is automorphic for every set Γ. It remains open the question posed in [7] whether the only automorphic separable Banach spaces are ℓ 2 and c 0 and also the more general question whether all automorphic Banach spaces are isomorphic either to ℓ 2 (Γ) or to c 0 (Γ) for some set Γ. Our aim in this note is to address this latter problem for the case of Banach spaces C(K) of continuous functions on compact spaces. Must be an automorphic C(K) space isomorphic to c 0 (Γ)? We provide a positive answer to this problem in the case when K is a continuous image of a Valdivia compact, which is a large class of compact spaces originated from functional analyisis and which includes for example all Eberlein and all dyadic compact spaces. Namely, a compact space is said to be a Valdivia compact if it is homeomorphic to some K ⊂ R Γ in such a way that the elements of K of countable support are dense in K (the support of x ∈ R Γ is the set of nonzero coordinates). If such K can be found so that all elements of K have countable support, then the compact is said to be a Corson compact, and if moreover it can be taken so that K ⊂ c 0 (Γ) ⊂ R Γ , then it is called an Eberlein compact. 2000 Mathematics Subject Classification. 46B26. The first author was supported by a Marie Curie Intra-European Felloship MCEIF-CT2006- 038768 (E.U.) and research projects MTM2005-08379 and S´ eneca 00690/PI/04 (Spain). 1