(Non-)Dunford-Pettis operators on noncommutative symmetric spaces
An operator \(T\) from a Banach space \(X \) into another Banach space \(Y\) is called a Dunford-Pettis operator (shortly, (DP)-operator), whenever $T$ maps weakly null sequences into norm null sequences. A Banach space $X$ has the Dunford-Pettis property (shortly, DPP) if all weakly compact operators defined from $X$ into an arbitrary Banach space $Y$ are Dunford-Pettis operators, or equivalently, if for all weakly null sequences \((x_n)\) in \(X\) and \((f_n)\) in \(X^*\), we have \(f_n(x_n) \to 0\). The authors consider two important questions concerned with a von Neumann algebra. One asks: if an operator on the predual \(\mathcal M_*\) of a von Neumann algebra \(\mathcal M\) acting on a separable Hilbert space \(\mathcal H\) is a non-Dunford-Pettis operator, is there a subspace \(Z\) of \(\mathcal M_*\) such that \(Z\) is isomorphic to \(\bigoplus_{\ell_1}\ell_2\) with \(T|_Z\) an isomorphism? They give the following characterizations: Theorem 1. Let \(\mathcal M\) be a von Neumann algebra acting on a separable Hilbert space \(\mathcal H\); then the following statements are equivalent: \((1)\) \(\mathcal M=\bigoplus_{1\leq n<\infty}(\mathcal A_n\otimes B(\ell_2^n))\), where the\(\mathcal A_n\)'s are commutative von Neumann algebras and at least one of them is not atomic; \((2)\) \(\mathcal M_*\) is isomorphic to one of the following spaces \[ L_1(0,1),\ S_1\oplus L_1(0,1), \ L_1(S_1),\] where \(S_1\) is the \(\ell_1\)-sum of the preduals to \(B(\ell_2^n)\), \(n=1,2,\dots\); \((3)\) any non-(DP)-operator \(T\) on \(\mathcal M_*\) fixes a copy of \(\oplus_{\ell_1}(\ell_2)\). Another problem is: for which symmetric function spaces \(E(0,\infty)\) and semifinite von Neumann algebras \(\mathcal M\) (equipped with semifinite faithful normal traces \(\tau\)) do the noncommutative symmetric spaces \(E(\mathcal M, \tau)\) have the DPP? They answer this problem and fully characterize those noncommutative symmetric spaces affiliated with different types of von Neumann algebras which have the Dunford-Pettis property.
scientific article; zbMATH DE number 7495781
scientific article