Illinois Journal of Mathematics

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

N. J. Kalton

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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.

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Illinois J. Math., Volume 52, Number 1 (2008), 279-317.

First available in Project Euclid: 15 May 2009

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Zentralblatt MATH identifier

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B20: Geometry and structure of normed linear spaces


Kalton, N. J. Automorphisms of $\mathcal{C}(K)$-spaces and extension of linear operators. Illinois J. Math. 52 (2008), no. 1, 279--317. doi:10.1215/ijm/1242414132.

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