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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| Vydané v: | Canadian journal of mathematics Ročník 44; číslo 4; s. 797 - 804 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Cambridge, UK
Cambridge University Press
01.08.1992
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| ISSN: | 0008-414X, 1496-4279 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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 ƒ
n
→ 0 weakly in A* and
φn
→ 0 weakly in A* then
φn(ƒ
n
)
→ 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
B
b
,
introduced by [6] Cima and Timoney. The algebra
B
b
is the set of ƒ in A such that if ƒ
n
→ 0 weakly in B then dist(ƒƒ
n
,
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
X
b
— C(L).
Cima and Timoney proved that
Bb
is a closed subalgebra of A and that if
B
is an algebra then
B
⊂
B
b
.
In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle
T. 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. |
| Author | Gorkin, Pamela Izuchi, Keiji Mortini, Raymond |
| Author_xml | – sequence: 1 givenname: Pamela surname: Gorkin fullname: Gorkin, Pamela organization: Bucknell University, Lewisburg, Pennsylvania 17837, U.S.A – sequence: 2 givenname: Keiji surname: Izuchi fullname: Izuchi, Keiji organization: Kanagawa University, Rokkakubashi, Kanagawa, Yokohama 221, Japan – sequence: 3 givenname: Raymond surname: Mortini fullname: Mortini, Raymond organization: Mathematisches Institut I, Universität Karlsruhe, Englerstrasse 2, D-7500 Karlsruhe, Germany |
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| References | Hoffman (S0008414X00010841_ref010) 1962 Younis (S0008414X00010841_ref017) 1985; 15 S0008414X00010841_ref016 Sundberg (S0008414X00010841_ref015) 1983; 276 Cima (S0008414X00010841_ref006) 1987; 34 Newman (S0008414X00010841_ref013) 1959; 92 Hoffman (S0008414X00010841_ref011) 1967; 86 Bourgain (S0008414X00010841_ref002) 1984; 77 Cima (S0008414X00010841_ref005) 1989; 105 Marshall (S0008414X00010841_ref012) 1976; 137 Chang (S0008414X00010841_ref004) 1976; 137 Sarason (S0008414X00010841_ref014) 1978 Gorkin (S0008414X00010841_ref008) 1988; 104 Axler (S0008414X00010841_ref001) 1984; 31 S0008414X00010841_ref009 Budde (S0008414X00010841_ref003) 1982 Garnett (S0008414X00010841_ref007) 1981 |
| References_xml | – volume-title: Function theory on the unit circle year: 1978 ident: S0008414X00010841_ref014 – ident: S0008414X00010841_ref009 – volume: 31 start-page: 89 year: 1984 ident: S0008414X00010841_ref001 article-title: Michigan Math. J. publication-title: Divisibility in Douglas algebras – volume-title: Bounded analytic functions year: 1981 ident: S0008414X00010841_ref007 – volume-title: Support sets and Gleason parts ofM(H°°) year: 1982 ident: S0008414X00010841_ref003 – volume: 15 start-page: 555 year: 1985 ident: S0008414X00010841_ref017 article-title: Arch. Math. publication-title: Division in Douglas algebras and some applications – volume: 86 start-page: 74 year: 1967 ident: S0008414X00010841_ref011 article-title: Ann. of Math. (2) publication-title: Bounded analytic functions and Gleason parts – volume: 137 start-page: 91 year: 1976 ident: S0008414X00010841_ref012 article-title: Acta Math. publication-title: Subalgebrasof L∞ containing H∞ – volume: 77 start-page: 245 year: 1984 ident: S0008414X00010841_ref002 article-title: Studia Math. publication-title: The Dunford Pettis property for the ball algebras the polydisc algebras and Sobolev spaces, – ident: S0008414X00010841_ref016 – volume: 34 start-page: 99 year: 1987 ident: S0008414X00010841_ref006 article-title: Michigan Math. J. publication-title: The Dunford Pettis property for certain planar uniform algebras – volume: 105 start-page: 121 year: 1989 ident: S0008414X00010841_ref005 article-title: Proc. Amer. Math. Soc. publication-title: Completely continuous HankeI operators on H∞ and Bourgain algebras, – volume: 137 start-page: 81 year: 1976 ident: S0008414X00010841_ref004 article-title: Acta. Math. publication-title: characterization of Douglas algebras – volume-title: Banach spaces of analytic functions year: 1962 ident: S0008414X00010841_ref010 – volume: 276 start-page: 551 year: 1983 ident: S0008414X00010841_ref015 article-title: Trans. Amer. Math. Soc. publication-title: Interpolating sequences for QAB – volume: 92 start-page: 501 year: 1959 ident: S0008414X00010841_ref013 article-title: Trans. Amer. Math. Soc. publication-title: Interpolation in H∞ – volume: 104 start-page: 1086 year: 1988 ident: S0008414X00010841_ref008 article-title: Proc. Amer. Math. Soc. publication-title: Functions not vanishing on trivial Gleason parts of Douglas algebras |
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