Sets of uniformly absolutely continuous norm in symmetric spaces of measurable operators (Q2790624)

Let \((\mathcal{M},\tau)\) be a von Neumann algebra with a semi-finite faithful normal trace \(\tau\). Let \(E\) be a strongly symmetric space of \(\tau\)-measurable operators affiliated with \(\mathcal{M}\). Such spaces \(E\) are considered as non-commutative analogues of rearrangement invariant function spaces. (The authors point out that their terminology differs from the one in two former papers.) Recall that the norm on \(E\) is called order continuous if \(\|x_\alpha\|\searrow0\) whenever \(x_\alpha\searrow0\) in \(E\).NEWLINENEWLINEA bounded subset \(\mathcal{A}\) of \(E\) is said to be of uniformly absolutely continuous (uac for short) norm if \(\|e_nxe_n\|\) converges to \(0\) uniformly in \(x\in\mathcal{A}\) for any sequence \((e_n)\) of decreasing projections of \(\mathcal{A}\) with infimum \(0\).NEWLINENEWLINEThe authors' motivation is Akemann's classical result that for bounded subsets of the predual of a von Neumann algebra -- in particular of \(L^1(\tau)=\mathcal{M}_*\) -- relative weak compactness and being of uac norm coalesce [\textit{C. A. Akemann}, Trans. Am. Math. Soc. 126, 286--302 (1967; Zbl 0157.44603)]. This result, in turn, is reminiscent of Dunford and Pettis' well-known characterization of equiintegrable bounded subsets of \(L^1\); in fact, if \(E=L^p(\tau)\) (\(1\leq p<\infty\)), then being of uac norm is the same as equiintegrability as defined, for example, in [\textit{N. Randrianantoanina}, J. Oper. Theory 48, No. 2, 255--272 (2002; Zbl 1029.46100)] or in [\textit{Y. Raynaud} and \textit{Q.-H. Xu}, J. Funct. Anal. 203, No. 1, 149--196 (2003; Zbl 1056.46056)].NEWLINENEWLINEAfter proving a useful technical characterization of sets of uac norm, the authors show that one implication of Akemann's result holds for all strongly symmetric \(E\) with order continuous norm: in such spaces, bounded sets with uac norm are relatively weakly compact. In general, the converse fails but the authors give several conditions in terms of finiteness of \(\tau\) or in terms of the measure topology for the converse to hold. As a by-product, it is shown that if \(E\) contains an isomorphic copy of \(c_0\), then it contains one generated by a sequence of pairwise orthogonal positive elements.NEWLINENEWLINECharacterizations of compactness of bounded subsets of certain \(E\)'s are given, too.NEWLINENEWLINESome `non-commutative' Lorentz and Orlicz spaces are shown to have the property (well-known for \(L^1\)) that a sequence converging to \(0\) both weakly and in measure converges in norm.NEWLINENEWLINEIn the last sections, the authors address the situation of two completely positive operators \(T, S\) between two strongly symmetric spaces \(E\) and \(F\) such that \(T\) completely dominates \(S\) in the sense that \(T-S\) is completely positive (to resume: \(0\leq_{cp}S\leq_{cp}T\)), and they consider the question when certain properties of \(T\) pass to \(S\). For example, if \(E=F\) is a finite von Neumann algebra or one of the aforementioned Lorentz or Orlicz spaces, then \(S\) is Dunford-Pettis as soon as \(T\) is.