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Some Remarks on the Dunford-Pettis Property

1997; Rocky Mountain Mathematics Consortium; Volume: 27; Issue: 4 Linguagem: Inglês

10.1216/rmjm/1181071869

1945-3795

Narcisse Randrianantoanina,

Advanced Harmonic Analysis Research

Let A be the disk algebra, Ω be a compact Hausdorff space and µ be a Borel measure on Ω.It is shown that the dual of C(Ω, A) has the Dunford-Pettis property.This proved in particular that the spaces L 1 (µ, L 1 /H 1 0 ) and C(Ω, A) have the Dunford-Pettis property. Introduction.Let E be a Banach space, Ω be a compact Hausdorff space and µ be a finite Borel measure on Ω.We denote by C(Ω, E) the space of all E-valued continuous functions from Ω and for 1 ≤ p < ∞, L p (µ, E) stands for the space of all (class of) E-valued p-Bochner integrable functions with its usual norm.A Banach space E is said to have the Dunford-Pettis property if every weakly compact operator with domain E is completely continuous, i.e., takes weakly compact sets into norm compact subsets of the range space.There are several equivalent definitions.The basic result proved by Dunford and Pettis in [11] is that the space L 1 (µ) has the Dunford-Pettis property.A. Grothendieck [12] initiated the study of Dunford-Pettis property in Banach spaces and showed that C(K)-spaces have this property.The Dunford-Pettis property has a rich history; the survey articles by J. Diestel [8] and A. Pe lczyński [15] are excellent sources of information.In [8] it was asked if the Dunford-Pettis property can be lifted from a Banach E to C(Ω, E) or L 1 (µ, E).M. Talagrand [18] constructed counterexamples for these questions so the answer is negative in general.There are, however, some positive results.For instance, J. Bourgain showed (among other things) in [2] that C(Ω, L 1 ) and L 1 (µ, C(Ω)) both have the Dunford-Pettis property; K. Andrews [1] proved that if E * has the Schur property then L 1 (µ, E) has the Dunford-Pettis property.F. Delbaen [7] showed that if A is the disc algebra, then L 1 (µ, A) has the Dunford-Pettis property.In [17], E. Saab and P. Saab observed that if A is a C * -algebra with the Dunford-Pettis property then C(Ω, A)