The alternative Dunford-Pettis property in \(C^*\)-algebras and von Neumann preduals. (English) Zbl 1020.46003
A Banach space \(E\) is said to have the Dunford-Pettis property if, for each pair \((x_j)\) and \((\rho_j)\) of sequences in \(E\) and its dual space \(E^*\), respectively, both of which converge weakly to zero, the sequence \((\rho_j(x_j))\) converges to zero. A Banach space \(E\) with the Dunford-Pettis property automatically has the alternative Dunford-Pettis property, that, if \((x_j)\) is a sequence of elements of norm one in \(E\) converging weakly to an element \(x\) in \(E\) of norm one, and \((\rho_j)\) is a sequence in \(E^*\) converging weakly to zero, then the sequence \((\rho_j(x_j))\) converges to zero. The authors show that, when \(E\) is a \(C^*\)-algebra, the two conditions are equivalent, and that the predual of a \(W^*\)-algebra \(A\) has the alternative Dunford-Pettis property if and only if \(A\) is of Type I.
Reviewer: C.M.Edwards (Oxford)
MSC:
| 46B28 | Spaces of operators; tensor products; approximation properties |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L10 | General theory of von Neumann algebras |
Keywords:
\(C^*\)-algebra; \(W^*\)-algebra; Dunford-Pettis property; alternative Dunford-Pettis propertyReferences:
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