The unit ball of the predual of \(H^\infty (\mathbb {B}_d)\) has no extreme points (Q2790265)

Let \(H^\infty(\mathbb{B}_d)\) denote the algebra of bounded analytic functions on the open unit ball \(\mathbb{B}_d \subset \mathbb{C}^d\). This space has a predual, that is, a Banach space \(H^\infty(\mathbb{B}_d)_*\) such that the dual \((H^\infty(\mathbb{B}_d)_*)^*\) is isometrically isomorphic to \(H^\infty(\mathbb{B}_d)\). In this paper, the authors consider the predual \(H^\infty(\mathbb{B}_d)_*= L^\infty(\partial \mathbb{B}_d)/ H^\infty(\mathbb{B}_d)_{\perp}\) (it is not known whether this is the unique predual). The authors show that the unit ball of this predual has no extreme points (the case \(d=1\) was proved by \textit{T. Ando} [Commentat. Math., 33--40 (1978; Zbl 0384.46035)]).NEWLINENEWLINEFor their proof, the authors calculate the extreme points and the so-called \textit{weak-\(*\) exposed} points of the unit ball of \(A(\mathbb{B}_d)^*\) (the dual of the ball algebra), and then apply a theorem that they attribute to Klee, which is a Krein-Milman type result with weak-\(*\) exposed points replacing extreme points (the authors provide the proof of Klee's theorem, since apparently it does not appear in the literature).NEWLINENEWLINEThe authors note that the idea of this paper arose in connection with another work of theirs, [Adv. Math. 295, 90--149 (2016; Zbl 1422.47073)], which treats related problems in the setting of multipliers on Drury-Arveson spaces.