The unit ball of the predual of 𝐻^{∞}(𝔹_{𝕕}) has no extreme points

Clouâtre, Raphaël; Davidson, Kenneth

doi:10.1090/proc/12964

The unit ball of the predual of $H^\infty (\mathbb {B}_d)$ has no extreme points
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by Raphaël Clouâtre and Kenneth R. Davidson

Proc. Amer. Math. Soc. 144 (2016), 1575-1580

DOI: https://doi.org/10.1090/proc/12964

Published electronically: November 20, 2015

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Abstract:

We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of $H^\infty (\mathbb {B}_d)$ has no extreme points in its unit ball.

References

T. Ando, On the predual of $H^{\infty }$, Comment. Math. Spec. Issue 1 (1978), 33–40. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 504151
R. Clouâtre and K. Davidson, Duality, convexity and peak interpolation in the Drury-Arveson space, preprint arXiv:1504.00665, 2015.
Brian Cole and R. Michael Range, $A$-measures on complex manifolds and some applications, J. Functional Analysis 11 (1972), 393–400. MR 0340646, DOI 10.1016/0022-1236(72)90061-4
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Kenneth R. Davidson and Alex Wright, Operator algebras with unique preduals, Canad. Math. Bull. 54 (2011), no. 3, 411–421. MR 2857916, DOI 10.4153/CMB-2011-036-0
I. Glicksberg, The abstract F. and M. Riesz theorem, J. Functional Analysis 1 (1967), 109–122. MR 0211264, DOI 10.1016/0022-1236(67)90029-8
G. M. Henkin, The Banach spaces of analytic functions in a ball and in a bicylinder are nonisomorphic, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 82–91 (Russian). MR 0415288
Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR 0410335
Matthew Kennedy and Dilian Yang, A non-self-adjoint Lebesgue decomposition, Anal. PDE 7 (2014), no. 2, 497–512. MR 3218817, DOI 10.2140/apde.2014.7.497
V. L. Klee Jr., Extremal structure of convex sets. II, Math. Z. 69 (1958), 90–104. MR 92113, DOI 10.1007/BF01187394
Heinz König and G. L. Seever, The abstract F. and M. Riesz theorem, Duke Math. J. 36 (1969), 791–797. MR 287319
S. Mazur, Uber konvexe Mengen in linearen normieren Räumem, Studia Math. 4 (1933), 70–84.
D. Mil′man, Accessible points of a functional compact set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 1045–1048 (Russian). MR 0024075
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
R. È. Val′skiĭ, Measures that are orthogonal to analytic functions in $C^{n}$, Dokl. Akad. Nauk SSSR 198 (1971), 502–505 (Russian). MR 0285899
Kôsaku Yosida and Masanori Fukamiya, On regularly convex sets, Proc. Imp. Acad. Tokyo 17 (1941), 49–52. MR 4069