Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quotients of $ L\sp{\infty }$ by Douglas algebras and best approximation


Authors: Daniel H. Luecking and Rahman M. Younis
Journal: Trans. Amer. Math. Soc. 276 (1983), 699-706
MSC: Primary 46J15; Secondary 30H05
DOI: https://doi.org/10.1090/S0002-9947-1983-0688971-4
MathSciNet review: 688971
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that $ {L^\infty}/A$ is not the dual space of any Banach space when $ A$ is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in $ {L^\infty}$ has a nonunique best approximation in $ A$. We therefore also show that the Douglas algebra $ {H^\infty} + L_F^\infty $, when $ F$ is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi [6] and independently obtained by Marshall and Zame.


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

  • [1] S. Axler, Factorization of $ {L^\infty}$ functions, Ann. of Math. (2) 106 (1977), 567-572. MR 0461142 (57:1127)
  • [2] S. Axler, I. D. Berg, N. Jewell and A. L. Shields, Approximation by compact operators and the space $ {H^\infty} + C$, Ann. of Math. (2) 109 (1979), 601-612. MR 534765 (81h:30053)
  • [3] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547-559. MR 0141789 (25:5186)
  • [4] S.-Y. A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), 81-89. MR 0428044 (55:1074a)
  • [5] T. W. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, N. J., 1969. MR 0410387 (53:14137)
  • [6] E. Hayashi, The spectral density of a strongly mixing stationary Gaussian process, preprint. MR 637976 (83b:46034)
  • [7] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 0133008 (24:A2844)
  • [8] -, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74-111. MR 0215102 (35:5945)
  • [9] R. Holmes, B. Scranton and J. Ward, Approximation from the space of compact operators and other $ M$-ideals, Duke Math. J. 42 (1975), 259-269. MR 0394301 (52:15104)
  • [10] D. H. Luecking, The compact Hankel operators form an $ M$-ideal in the space of Hankel operators, Proc. Amer. Math. Soc. 79 (1980), 222-224. MR 565343 (81h:46057)
  • [11] D. E. Marshall, Subalgebras of $ {L^\infty}$ containing $ {H^\infty}$, Acta Math. 137 (1976), 91-98. MR 0428045 (55:1074b)
  • [12] D. Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286-299. MR 0324425 (48:2777)
  • [13] -, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc, 207 (1975), 391-405. MR 0377518 (51:13690)
  • [14] -, Function theory on the unit circle, Lecture notes for a conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19-23, 1978. MR 521811 (80d:30035)
  • [15] R. Younis, Best approximation in certain Douglas algebras, Proc. Amer. Math. Soc. 80 (1980), 639-642. MR 587943 (81j:46031)
  • [16] -, Properties of certain algebras between $ {L^\infty}$ and $ {H^\infty}$, J. Funct. Anal. 44 (1981), 381-387. MR 643041 (83b:46071)
  • [17] -, Extension results in the Hardy space associated with a logmodular algebra, J. Funct. Anal. 39 (1980), 16-22. MR 593785 (82a:46052)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46J15, 30H05

Retrieve articles in all journals with MSC: 46J15, 30H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0688971-4
Keywords: Douglas algebras, dual space, best approximation
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society