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Orthogonally additive polynomials over \(C(K)\) are measures. A short proof. (English) Zbl 1122.46025

A \(k\)-homogeneous polynomial \(P:C(K) \to \mathbb K\) is called orthogonally additive if \(P(u + v) = P(u) + P(v)\) whenever \(u, v \in C(K)\) and \(uv = 0.\) The authors provide the third and simplest proof of the following theorem: For such a polynomial \(P\) on \(C(K),\) there is a regular Borel measure \(\mu\) on \(K\) such that \(P(u) = \int_K u^k \,d\mu\) for every \(u \in C(K).\) Earlier demonstrations of this result were given by D.Pérez-García and I.Villanueva [J. Math.Anal.Appl.306, No.1, 97–105 (2005; Zbl 1076.46035)] and by Y.Benyamini, S.Lassalle, and J.G.Llavona [Bull.Lond.Math.Soc.38, No.3, 459–469 (2006; Zbl 1110.46033)]. The ingredients of the present argument include a linearization result due to D.Carando and I.Zalduendo [Proc.Am.Math.Soc.127, 241–250 (1999; Zbl 0908.46031)] and the canonical result about the extension of a polynomial to the bidual [R.Aron and P.Berner, Bull.Soc.Math.Fr.106, 3–24 (1978; Zbl 0378.46043)].

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
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