Bourgain Algebras of Douglas Algebras
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...
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| Veröffentlicht in: | Canadian journal of mathematics Jg. 44; H. 4; S. 797 - 804 |
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| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge, UK
Cambridge University Press
01.08.1992
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| Schlagworte: | |
| ISSN: | 0008-414X, 1496-4279 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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. |
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| ISSN: | 0008-414X 1496-4279 |
| DOI: | 10.4153/CJM-1992-047-6 |