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