Multiplicative functionals on algebras of differentiable functions

Resumen

Let O be an open subset of a real Banach space E and, for 1 = m =, let Cm(O) denote the algebra of all m-times continuously Fréchet differentiable real functions defined on O. We are concerned here with the question as to wether every nonzero algebra homomorphism f: Cm(O) ? R is given by evaluation at some point of O, i.e., if there exists some a Î O such that f(f) = f(a) for each f Î Cm(O). This problem has been considered in [1,4,5] and [6]. In [6], a positive answer is given in the case that m < 8 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(O). In [1] (respectively, in [4]) a positive answer is given in the case that O = 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].