Automorphisms of $\mathcal{C}(K)$-spaces and extension of linear operators

Abstract

We study the class of separable (real) Banach spaces $X$ which can be embedded into a space $\mathcal{C}(K)$ ($K$ compact metric) in only one way up to automorphism. We show that in addition to the known spaces $c_0$ (and all it subspaces) and $ℓ_1$ (and all its weak^*-closed subspaces) the space $c_{0}(ℓ_{1})$ has this property. We show on the other hand (answering a question of Castillo and Moreno) that $ℓ_p$ for $1 < p < ∞$ fails this property. We also show that $ℓ_p$ can be embedded in a super-reflexive space $X$ so that there is an operator $T : \ell_{p}\to\mathcal{C}(K)$ which has no extension, answering a question of Zippin.