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Bourgain Algebras of Douglas Algebras

Published online by Cambridge University Press:  20 November 2018

Pamela Gorkin ,
Keiji Izuchi  and
Raymond Mortini
Pamela Gorkin
Affiliation:
Bucknell University, Lewisburg, Pennsylvania17837, U.S.A.
Keiji Izuchi
Affiliation:
Kanagawa University, Rokkakubashi, Kanagawa, Yokohama221, Japan
Raymond Mortini
Affiliation:
Mathematisches Institut I, Universität Karlsruhe, Englerstrasse 2, D-7500 Karlsruhe, Germany
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Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.

Keywords

46J1046J30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 797 - 804
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Axler, S. and Gorkin, P., Divisibility in Douglas algebras, Michigan Math. J. 31 (1984), 89–94.Google Scholar
2. Bourgain, J., The Dunford Pettis property for the ball algebras the polydisc algebras and Sobolev spaces, Studia Math. 77 (1984), 245–253.Google Scholar
3. Budde, P., Support sets and Gleason parts ofM(H°°), Ph. thesis, D., Univ. of California, Berkely, 1982.Google Scholar
4. Chang, S.-Y., A characterization of Douglas algebras, Acta. Math. 137 (1976), 81–89.Google Scholar
5. Cima, J., Janson, S. and Yale, K., Completely continuous HankeI operators on H∞ and Bourgain algebras, Proc. Amer. Math. Soc. 105 (1989), 121–125.Google Scholar
6. Cima, J. and Timoney, R., The Dunford Pettis property for certain planar uniform algebras, Michigan Math. J. 34 (1987), 99–104.Google Scholar
7. Garnett, J., Bounded analytic functions, Academic Press, New York and London, 1981.Google Scholar
8. Gorkin, P., Functions not vanishing on trivial Gleason parts of Douglas algebras, Proc. Amer. Math. Soc. 104 (1988), 1086–1090.Google Scholar
9. Guillory, C., Izuchi, K. and Sarason, D., Interpolating Blaschke products and division in Douglas algebras, Proc. Royal Irish Acad. 84A(1984), 1–7.Google Scholar
10. Hoffman, K., Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N.J., 1962.Google Scholar
11. Hoffman, K., Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111.Google Scholar
12. Marshall, D., Subalgebrasof L∞ containing H∞ , Acta Math. 137 (1976), 91–98.Google Scholar
13. Newman, D.J., Interpolation in H∞ , Trans. Amer. Math. Soc. 92 (1959), 501–507.Google Scholar
14. Sarason, D., Function theory on the unit circle, Virginia Polytechnic Inst, and State Univ., Blacksburg, 1978.Google Scholar
15. Sundberg, C. and Wolff, T., Interpolating sequences for QAB, Trans. Amer. Math. Soc. 276 (1983), 551–581.Google Scholar
16. Yale, K., Bourgain algebras, preprint.Google Scholar
17. Younis, R., Division in Douglas algebras and some applications, Arch. Math. 15 (1985), 555–560.Google Scholar