Many authors have recently studied compact and weakly compact homomorphisms between function algebras. Among them, Lindström and Llavona [2] treat weakly compact continuous homomorphisms between algebras of type C(T) when T is a completely regular Hausdorff space.
Llavona asked wether the results in [2] are valid in the case of algebras of differentiable functions on Banach spaces. The purpose of this note is to give an affirmative answer to this question, by proving that weakly compact homomorphisms between algebras of differentiable functions are induced by constant mappings. The difficulty we face is that in [2] the existence of continuous functions separating points and closed sets plays an essential role, while in the differentiable case these functions do not exist in general. We deal with Fréchet differentiability, but our results are also valid for Hadamard differentiable functions.
@article{urn:eudml:doc:39886,
title = {Compact and weakly compact homomorphisms between algebras of differentiable functions.},
journal = {Extracta Mathematicae},
volume = {5},
year = {1990},
pages = {118-120},
zbl = {0743.46019},
mrnumber = {MR1125678},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39886}
}
González, Manuel; Gutiérrez, Joaquín M. Compact and weakly compact homomorphisms between algebras of differentiable functions.. Extracta Mathematicae, Tome 5 (1990) pp. 118-120. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39886/