# Multiplicative functionals on algebras of differentiable functions.

Extracta Mathematicae (1990)

• Volume: 5, Issue: 3, page 144-146
• ISSN: 0213-8743

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## Abstract

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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 &lt; ∞ 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].

## How to cite

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Jaramillo, Jesús A.. "Multiplicative functionals on algebras of differentiable functions.." Extracta Mathematicae 5.3 (1990): 144-146. <http://eudml.org/doc/39896>.

@article{Jaramillo1990,
abstract = {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 &lt; ∞ 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].},
author = {Jaramillo, Jesús A.},
journal = {Extracta Mathematicae},
keywords = {Algebra de funciones; Anillos de funciones; Funcional lineal; Multiplicadores; Homomorfismos; Algebra topológica; homomorphism; point evaluation; algebras of differentiable functions; super-reflexive Banach spaces with nonmeasurable cardinal},
language = {eng},
number = {3},
pages = {144-146},
title = {Multiplicative functionals on algebras of differentiable functions.},
url = {http://eudml.org/doc/39896},
volume = {5},
year = {1990},
}

TY - JOUR
AU - Jaramillo, Jesús A.
TI - Multiplicative functionals on algebras of differentiable functions.
JO - Extracta Mathematicae
PY - 1990
VL - 5
IS - 3
SP - 144
EP - 146
AB - 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 &lt; ∞ 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].
LA - eng
KW - Algebra de funciones; Anillos de funciones; Funcional lineal; Multiplicadores; Homomorfismos; Algebra topológica; homomorphism; point evaluation; algebras of differentiable functions; super-reflexive Banach spaces with nonmeasurable cardinal
UR - http://eudml.org/doc/39896
ER -

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