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Non-commutative analogues of weak compactness criteria in symmetric spaces. (English) Zbl 1478.46057

The weak compactness criteria alluded to in the title go back to the following three ones in \(L^1(\nu)\)-spaces over a finite measure \(\nu\), to wit
Dunford-Pettis’ well-known classical equivalence of weak compactness and equi-integrability (for bounded sets);
De la Vallée Poussin’s equivalence of weak compactness and the existence of an \(N\)-function \(G\) (which is an Orlicz function satisfying some additional continuity conditions) such that \(\sup\{\int G(|f|)\}<\infty\), where the \(\sup\) runs over all \(f\) belonging to the set in question;
Chong’s description of weakly compact sets as exactly those which are “in the orbit” of an appropriate positive integrable function \(g\), more precisely, which are contained in the set \(\{f : |f|\prec\prec g\}\) where the symbol \(f\prec\prec g\) stands for the condition \(\int_0^t\mu(f)\le\int_0^t\mu(g)\) for all \(t\) with \(\mu(f)\) being the decreasing rearrangement of \(|f|\).

There is a further criterion of T. Ando [Can. J. Math. 14, 170–176 (1962; Zbl 0103.32902)] concerning weak compactness with respect to Köthe duality in Orlicz spaces \(L_G\). (In some cases, this topology coincides with the usual (Banach space) weak topology considered in the article under review.)
The setting of the paper is the one of non-commutative Orlicz spaces \(\mathcal{L}_G(\mathcal{M},\tau)\) (in particular, non-commutative Lebesgue spaces \(\mathcal{L}_p(\mathcal{M})\)) and of non-commutative symmetric operator spaces \(\mathcal{E}=\mathcal{E}(\mathcal{M},\tau)\), both associated in the natural way to a von Neumann algebra \(\mathcal{M}\) with a semifinite faithful normal trace \(\tau\). The authors generalize known results on weak compactness criteria to this context.
They show, for example, an analogue of Ando’s criterion for non-commutative Orlicz spaces. They also show the equivalence of De la Vallée Poussin’s and Chong’s criteria for non-commutative \(\mathcal{L}_1(\mathcal{M})\) where \(\mathcal{M}\) is non-atomic finite, more precisely, for a bounded subset \(\mathcal{K}\) of \(\mathcal{L}_1(\mathcal{M})\) there exists an Orlicz function \(G\) with \(G(t)/t\to\infty\) as \(t\to\infty\) and \(\sup_{A\in\mathcal{K}}\tau(G(|A|))<\infty\) if and only if there exists a positive operator \(B\in\mathcal{L}_1(\mathcal{M})\) such that \(\mu(|A|)\prec\prec\mu(B)\) for all \(A\in\mathcal{K}\). (Here, \(\tau(G(|A|))\) is defined in the natural way so that it equals \(\int G(|A|)\) in the case where \(\mathcal{M}\) is the classical \(L^\infty\), and \(\mu(|A|)\) is the singular value function of \(|A|\) which generalizes the rearrangement function in the case \(\mathcal{M}=L^\infty\).)
In another direction. the authors study weak compactness in the context of Pełczynski’s property (V). \(M\)-embedded Banach spaces have this property. (A Banach space \(X\) is \(M\)-embedded if \(X^{***}=X^*\oplus_1 X^{\perp}\) where \(X^{\perp}=\{x^{***}\in X^{***}\,:\,x^{***}_{|X}=0\}\).) For classical Orlicz spaces it is known that \(H_G(0,\infty)=\{f\,:\,\int_0^\infty G(|f|/\rho)<\infty\), \(\forall\rho>0\}\) is \(M\)-embedded in \(L_G\) if the conjugate function \(G^*\) of \(G\) satisfies the \(\Delta_2\)-condition while \(G\) does not. Now the authors show that the corresponding non-commutative Orlicz spaces \(\mathcal{H}_G\) are \(M\)-embedded, too, and thus have property (V).
Finally, there is a criterion for compactness: Let \(\mathcal{F}\) be a bounded subset of a separable symmetric Banach space \(\mathcal{E}(\mathcal{M})\), where \(\mathcal{M}\) is hyperfinite, such that it is the \(w^*\)-closure of countably many increasing finite dimensional subalgebras \(\mathcal{M}_n\) with conditional expectations \(\mathbb{E}_n\). Then \(\mathcal{F}\) is relatively weakly compact if and only if \(\sup_{x\in\mathcal{F}}\|x-\mathbb{E}_nx\|_{\mathcal{E}(\mathcal{M})}\to0\).

MSC:

46L52 Noncommutative function spaces
46B50 Compactness in Banach (or normed) spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46L10 General theory of von Neumann algebras
46L51 Noncommutative measure and integration
54D30 Compactness
47L10 Algebras of operators on Banach spaces and other topological linear spaces
46B04 Isometric theory of Banach spaces
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