Abstract
Let H be a two-dimensional complex Hilbert space and \({{\mathcal P}(^3H)}\) the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, \({{\mathsf B}_{{\mathcal P}(^3H)}}\) , from which we deduce that the unit sphere of \({{\mathcal P}(^3H)}\) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of \({{\mathsf B}_{{\mathcal P}(^3H)}}\) remains extreme as considered as an element of \({{\mathsf B}_{{\mathcal L}(^3H)}}\) . Finally we make a few remarks about the geometry of the unit ball of the predual of \({{\mathcal P}(^3H)}\) and give a characterization of its smooth points.
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The writing of this article started while B. C. Grecu was a postdoctoral fellow at Departamento de Analisis Matematico, Universidad de Valencia. He was supported by a Marie Curie Intra European Fellowship (MEIF-CT-2005-006958).
G. A. Muñoz-Fernández and J. B. Seoane-Sepúlveda were supported by MTM 2006–03531.
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Grecu, B.C., Muñoz-Fernández, G.A. & Seoane-Sepúlveda, J.B. The unit ball of the complex \({{\mathcal P}(^3H)}\) . Math. Z. 263, 775–785 (2009). https://doi.org/10.1007/s00209-008-0438-y
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DOI: https://doi.org/10.1007/s00209-008-0438-y