Multiplicative functionals on algebras of differentiable functions.

Jaramillo, Jesús A.

Extracta Mathematicae, Tome 5 (1990), p. 144-146 / Harvested from Biblioteca Digital de Matemáticas

Résumé

Let Ω be an open subset of a real Banach space E and, for 1 ≤ m ≤, let Cm(Ω) denote the algebra of all m-times continuously Fréchet differentiable real functions defined on Ω. We are concerned here with the question as to wether every nonzero algebra homomorphism φ: Cm(Ω) → R is given by evaluation at some point of Ω, i.e., if there exists some a ∈ Ω such that φ(f) = f(a) for each f ∈ Cm(Ω). This problem has been considered in [1,4,5] and [6]. In [6], a positive answer is given in the case that m < ∞ and E is a Banach space which admits Cm-partitions of unity and with nonmeasurable cardinal; this result is obtained there as a by-product of the study of two topologies, and , introduces on Cm(Ω). In [1] (respectively, in [4]) a positive answer is given in the case that Ω = E is a separable Banach space (respectively, the dual of a separable Banach space). In the present note we extend these previous results, and we give an affirmative answer for a wider class of Banach spaces, including super-reflexive spaces with nonmeasurable cardinal. We also provide a direct approach and a unified treatment, since our results here are derived as a consequence of Theorem 1 below, a general result slightly in the spirit of Theorem 12.5 of [7].

Détail

Publié le : 1990-01-01

MR MR1125687 | Zbl 0801.46024

DMLE-ID : 2570