Geometry of Spaces of Orthogonally Additive Polynomials on C ( K )
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Geometry of Spaces of Orthogonally Additive Polynomials on C(K ) Christopher Boyd1 · Raymond A. Ryan2 · Nina Snigireva2 Received: 10 September 2018 © Mathematica Josephina, Inc. 2019
Abstract We study the space of orthogonally additive n-homogeneous polynomials on C(K ). There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive nhomogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach–Stone theorem. We conclude with a classification of the exposed points. Keywords Orthogonally additive · Homogeneous polynomial · Banach lattice · Regular polynomial · Extreme point · Exposed point · Isometry Mathematics Subject Classification 46G25 · 46G20 · 46E10 · 46B42 · 46B04 · 46E27
1 Introduction A real function f on a Banach lattice is said to be orthogonally additive if f (x + y) = f (x)+ f (y) whenever x and y are disjoint. Non-linear orthogonally additive functions on function spaces often have useful integral representations—see, for example, the papers of Chacon, Friedman and Katz [8,12,13], Mizel [21] and Rao [24]. In 1990,
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Raymond A. Ryan [email protected] Christopher Boyd [email protected] Nina Snigireva [email protected]
1
School of Mathematics & Statistics, University College Dublin, Belfield, Dublin 4, Ireland
2
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland
123
C. Boyd et al.
Sundaresan [26] initiated the study of orthogonally additive n-homogeneous polynomials with particular reference to the spaces L p [0, 1] and p for 1 ≤ p < ∞. Building on the work of Mizel, he showed that, for every orthogonally additive n-homogeneous polynomial P on L p [0, 1] with n ≤ p, there exists a unique function ξ ∈ L p˜ [0, 1], where p˜ = p/( p − n), such that
1
P(x) =
ξ x n dμ
(1)
0
for every x ∈ L p [0, 1]. When n > p, there are no non-zero orthogonally additive n-homogeneous polynomials on L p [0, 1]. He went on to show that the Banach space of orthogonally additive n-homogeneous polynomials on L p [0, 1] is isometrically isomorphic to L p˜ [0, 1] where the latter space is equipped not with the usual norm, but with the equivalent norm x = max{x + p˜ , x − p˜ }. The next significant development was the discovery of an integral representation for orthogonally additive n-homogeneous polynomials on C(K ) spaces by Pérez and Villanueva [23] and by Benyamini et al. [3], who proved a representation of the form P(x) =
x n dμ,
(2)
K
where μ is a regular Borel signed measure on K . The integral representations (1) and (2) have been extended and generalised in various directions in recent years. See, for example, [1,17,22,30]. Orth
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