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Bourgain Algebras of Douglas Algebras

Published online by Cambridge University Press:  20 November 2018

Pamela Gorkin
Bucknell University, Lewisburg, Pennsylvania17837, U.S.A.
Keiji Izuchi
Kanagawa University, Rokkakubashi, Kanagawa, Yokohama221, Japan
Raymond Mortini
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 BBb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.


Research Article
Copyright © Canadian Mathematical Society 1992


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