The Alternative Dunford-pettis Property in C∗-algebras and Von Neumann Preduals

A Banach space X is said to have the alternative Dunford-Pettis property if, whenever a sequence xn → x weakly in X with ‖xn‖ → ‖x‖, we have ρn(xn) → 0 for each weakly null sequence (ρn) in X∗. We show that a C∗-algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for C∗-algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I. Amongst several characterisations (see [6]), a convenient formulation of the Dunford-Pettis property (DP) in Banach spaces is that a Banach space E is said to have the DP if and only if whenever (xn) and (ρn) are weakly null sequences in E and E∗, respectively, then ρn(xn) → 0. The reader is referred to [6] for a comprehensive survey of the Dunford-Pettis property. All C(Ω)-spaces, alias commutative C∗-algebras, have the DP [8]. Subsequent investigations of general C∗-algebras in [3, 4] and [11] culminate in the proof [4] that a C∗-algebra has the DP if and only if it has no infinite dimensional irreducible representations. By [3] and [11] together with the observation [2], for the predual of a von Neumann algebra M to have the DP it is both necessary and sufficient that M be type I finite. In an interesting development, [7] introduced and studied the alternative Dunford-Pettis property, known in abbreviation as the DP1, referred to above. We recall that a Banach space E is said to have the DP1 if whenever a sequence xn → x weakly in E with ‖xn‖ = ‖x‖ = 1, for all n ∈ N, and (ρn) is a weakly null sequence in E∗, then ρn(xn) → 0. Several equivalent formulations of the DP1 are given in [7, 1.4] analogous to those of the DP given in [6, pages 17-18]. We remark that the DP1 is inherited by 1-complemented subspaces (cf. Remark 3). By confining the DP condition to the unit sphere of norm one elements, the DP1 allows greater freedom. In contrast to the DP, the DP1 is not isomorphism invariant [7, 1.6]. That the DP implies the DP1 is clear. The space of trace class operators on an infinite dimensional Hilbert space provides a counterexample to the opposite implication, [7, 2.4]. Nevertheless, by [7, 3.5], the DP and the DP1 Received by the editors September 27, 2001 and, in revised form, December 3, 2001. 2000 Mathematics Subject Classification. Primary 46B04, 46B20, 46L05, 46L10. The second author was partially supported by D.G.I.C.Y.T. project no. PB 98-1371, and Junta de Andalućıa grant FQM 0199. c ©2002 American Mathematical Society