Weak compactness criteria in symmetric spaces of measurable operators (Q2760487)

Let \({\mathcal M}\) be a semi-finite von Neumann algebra on the Hilbert space \(H\), with a fixed faithful and normal semifinite trace \(\tau\), \(\mathbf{1}\) the identity in \({\mathcal M}\), and \(\mu(x)\) the generalized singular value in the sense of \textit{T. Fack} and \textit{H. Kosaki} [Pac. J. Math. 123, 269-300 (1986; Zbl 0617.46063)].NEWLINENEWLINENEWLINELet \(E({\mathcal M},\tau)= \{x\in\widetilde M:\mu(x)\in E\}\), \(\|x\|_{E({\mathcal M},\tau)}= \|\mu(x)\|_E\) and \(E^\times({\mathcal M},\tau)\) be the noncommutative Köthe dual in the sense developed by the first named author, \textit{T. K.-Y. Dodds} and \textit{B. de Pagter} [Trans. Am. Math. Soc. 339, No. 2, 717-750 (1993; Zbl 0801.46074)]. The principal result of the paper is a simultaneous extension of the results of \textit{D. H. Fremlin} [Proc. Comb. Philos. Soc. 64, 625-643 (1968; Zbl 0187.07302)] and \textit{D. J. H. Garling} [Proc. Lond. Math. Soc., III. Ser. 17, 115-138 (1967; Zbl 0149.34202)] to the setting of symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra:NEWLINENEWLINENEWLINETheorem. Let \(E\subseteq L_0[0,\infty)\) be a rearrangement-invariant, fully symmetric Banach function space on \([0,\tau(\mathbf{1}))\), let \(K\subseteq E^\times({\mathcal M},\tau)\) be bounded. If \(({\mathcal M},\tau)\) is an arbitrary semi-finite von Neumann algebra, then the following statements are equivalent:NEWLINENEWLINENEWLINE(i) \(K\) is relatively \(\sigma(E^\times({\mathcal M},\tau),E({\mathcal M},\tau))\)-compact;NEWLINENEWLINENEWLINE(ii) \(\mu(K)\) is relatively \(\sigma(E^\times, E)\)-compact;NEWLINENEWLINENEWLINE(iii) \(\text{Orb}(\mu(K))\) is relatively \(\sigma(E^\times, E)\)-compact;NEWLINENEWLINENEWLINE(iv) \(\text{Orb}(K)= \{x\in L^1({\mathcal M},\tau)+{\mathcal M}: x\prec\prec y\) for some \(y\in K\}\)NEWLINENEWLINENEWLINEis relatively \(\sigma(E^\times({\mathcal M},\tau), E({\mathcal M},\tau))\)-compact, where \(x\prec\prec y\) denotes submajorization in the sense of Hardy-Littlewood-Pólya.