This is an announcement for the paper "Continuous multilinear functionals on $C(K)$-spaces are integral" by A. Ibort, P. Linares, and J. G. Llavona.

Abstract: In this paper we prove the theorem stated on the title: every continuous multilinear functional on $C(K)$-spaces is integral, or what is the same any polymeasure defined on the product of Borelian $\sigma$-algebras defined on compact sets can be extended to a bounded Borel measure on the compact product space. We provide two different proofs of the same result, each one stressing a different aspect of the various implications of this fact. The first one, valid for compact subsets of $\R^n$, is based on the classical multivariate theory of moments and is a natural extension of the Hausdorff moment problem to multilinear functionals. The second proof relies on a multilinear extension of the decomposition theorem of linear functionals on its positive and negative part which allows us prove a multilinear Riesz Theorem as well. These arguments are valid for arbitrary Hausdorff compact sets.

Archive classification: math.FA

Mathematics Subject Classification: 46G25

Remarks: 10 pages

The source file(s), Integralmultilinear.tex: 39365 bytes, is(are) stored in gzipped form as 0801.2878.gz with size 13kb. The corresponding postcript file has gzipped size 85kb.

Submitted from: plinares@mat.ucm.es

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