Vol. 65, No. 1, 1976

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The Dunford-Pettis property for certain uniform algebras

Freddy Delbaen

Vol. 65 (1976), No. 1, 29–33

A Banach space B has the Dunford-Pettis property if xn(xn) 0 whenever xn 0 weakly and the sequence xn tends to zero weakly in B (i.e. σ(B,B∗∗)). Suppose now that A is a uniform algebra on a compact space X. If ϕ is a nonzero multiplicative linear functional on A then Mϕ is the set of positive representing measures of ϕ. If A is such that a singular measure which is orthogonal to A must necessarily be zero and if all Mϕ are weakly compact sets then the algebra A as well as its dual have the Dunford-Pettis property.

Mathematical Subject Classification 2000
Primary: 46J10
Received: 4 August 1975
Revised: 20 November 1975
Published: 1 July 1976
Freddy Delbaen