The unit ball of the complex P(H-3)

Abstract

Let H be a two-dimensional complex Hilbert space and P(H-3) the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, B-P(3H), from which we deduce that the unit sphere of P(H-3) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B-P(3H) remains extreme as considered as an element of B-L(3H). Finally we make a few remarks about the geometry of the unit ball of the predual of P(H-3) and give a characterization of its smooth points.