Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: Raphaël Clouâtre and Kenneth R. Davidson
Journal: Proc. Amer. Math. Soc. 144 (2016), 1575-1580
MSC (2010): Primary 30H05, 46J15, 47L50
DOI: https://doi.org/10.1090/proc/12964
Published electronically: November 20, 2015
MathSciNet review: 3451234
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] T. Ando, On the predual of $ H^{\infty }$, Comment. Math. Special Issue 1 (1978), 33-40. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 504151 (80c:46063)
  • [2] R. Clouâtre and K. Davidson, Duality, convexity and peak interpolation in the Drury-Arveson space, preprint arXiv:1504.00665, 2015.
  • [3] Brian Cole and R. Michael Range, $ A$-measures on complex manifolds and some applications, J. Functional Analysis 11 (1972), 393-400. MR 0340646 (49 #5398)
  • [4] John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • [5] Kenneth R. Davidson and Alex Wright, Operator algebras with unique preduals, Canad. Math. Bull. 54 (2011), no. 3, 411-421. MR 2857916 (2012h:47153), https://doi.org/10.4153/CMB-2011-036-0
  • [6] I. Glicksberg, The abstract F. and M. Riesz theorem, J. Functional Analysis 1 (1967), 109-122. MR 0211264 (35 #2146)
  • [7] 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 (54 #3378)
  • [8] Richard B. Holmes, Geometric functional analysis and its applications, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 24. MR 0410335 (53 #14085)
  • [9] Matthew Kennedy and Dilian Yang, A non-self-adjoint Lebesgue decomposition, Anal. PDE 7 (2014), no. 2, 497-512. MR 3218817, https://doi.org/10.2140/apde.2014.7.497
  • [10] V. L. Klee Jr., Extremal structure of convex sets. II, Math. Z. 69 (1958), 90-104. MR 0092113 (19,1065b)
  • [11] Heinz König and G. L. Seever, The abstract F. and M. Riesz theorem, Duke Math. J. 36 (1969), 791-797. MR 0287319 (44 #4526)
  • [12] S. Mazur, Uber konvexe Mengen in linearen normieren Räumem, Studia Math. 4 (1933), 70-84.
  • [13] D. Milman, Accessible points of a functional compact set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 1045-1048 (Russian). MR 0024075 (9,449a)
  • [14] Walter Rudin, Function theory in the unit ball of $ {\bf C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594 (82i:32002)
  • [15] R. È. Valskiĭ, Measures that are orthogonal to analytic functions in $ C^{n}$, Dokl. Akad. Nauk SSSR 198 (1971), 502-505 (Russian). MR 0285899 (44 #3116)
  • [16] Kôsaku Yosida and Masanori Fukamiya, On regularly convex sets, Proc. Imp. Acad. Tokyo 17 (1941), 49-52. MR 0004069 (2,314b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30H05, 46J15, 47L50

Retrieve articles in all journals with MSC (2010): 30H05, 46J15, 47L50


Additional Information

Raphaël Clouâtre
Affiliation: Pure Mathematics Department, University of Waterloo, Waterloo, Ontario, Canada
Address at time of publication: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada
Email: raphael.clouatre@umanitoba.ca

Kenneth R. Davidson
Affiliation: Pure Mathematics Department, University of Waterloo, Waterloo, Ontario, Canada
Email: krdavids@uwaterloo.ca

DOI: https://doi.org/10.1090/proc/12964
Keywords: Bounded analytic functions, unit ball, extreme points, predual
Received by editor(s): April 3, 2015
Received by editor(s) in revised form: April 4, 2015
Published electronically: November 20, 2015
Additional Notes: The first author was partially supported by an FQRNT postdoctoral fellowship.
The second author was partially supported by an NSERC grant.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society