SPECTRAL SYNTHESIS AND OTHER RESULTS IN SOME TOPOLOGICAL ALGEBRAS OF VECTOR-VALUED FUNCTIONS

Abstract

Using results from theory of bundles of topological vector spaces, we prove some spectral synthesis and other results in certain topological algebras of vector-valued functions. In particular, we extend and generalize results of J.W. Kitchen and the second-named author (1996). We obtain as a corollary a recent (2008) result of Arizmendi-Peimbert/Carillo-Hoyo/Garc´ıa on the spectral synthesis property in (C_b(X); β), the space of bounded continuous functions on the completely regular Hausdorff space X under the strict topology β which arises from weighting by the non-negative upper semicontinuous functions on X which disappear at infinity.

Keywords: Spectral synthesis, bundles of topological vector spaces

Quaestiones Mathematicae 34(2011), 361–376