Quotients of 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

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Abstract: We show that is not the dual space of any Banach space when 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 has a nonunique best approximation in . We therefore also show that the Douglas algebra , when 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.

**[1]**S. Axler,*Factorization of**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*, 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*-*ideals*, Duke Math. J.**42**(1975), 259-269. MR**0394301 (52:15104)****[10]**D. H. Luecking,*The compact Hankel operators form an*-*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**containing*, 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**and*, 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)**

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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