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When weak and local measure convergence implies norm convergence. (English) Zbl 1423.46086
The paper addresses generalizations of the basic fact that, in \(L^1[0,1]\), a sequence converges in norm if (and only if) it converges simultaneously in the weak and in the measure topology. If the underlying measure \(\mu\) is only \(\sigma\)-finite, then the statement remains valid if convergence in measure is replaced by local measure convergence (i.e., convergence in measure on every set of finite measure).
The setting for the generalizations is as follows. Let \(\mathcal{M}\) be a von Neumann algebra equipped with a faithful normal semifinite trace \(\tau\) and consider \(S(\mathcal{M},\tau)\), the set of \(\tau\)-measurable operators affiliated with \(\mathcal{M}\). (If \(\mathcal{M}=L^\infty(\mu)\), then \(S(\mathcal{M},\tau)\) consists of all measurable functions which are bounded except on a set of finite measure.) On \(S(\mathcal{M},\tau)\), the (by now classical) complete metrizable measure topology \(t_\tau\) is considered: a sequence \((X_n)\) of \(S(\mathcal{M},\tau)\) converges in measure to \(0\) if \(\tau(\chi_{]\varepsilon,\infty[}(|X_n|))\to0\) for every \(\varepsilon>0\). Also, there is the weaker topology of local convergence in measure on \(S(\mathcal{M},\tau)\) which is defined naturally such that it corresponds to convergence in measure on sets of finite measure in the case \(\mathcal{M}=L^\infty(\mu)\). Elements \(X\) of \(\mathcal{M}\) with range projection of finite trace are said to be of \(\tau\)-finite range, their closure \(S_0(\mathcal{M},\tau)\) in \(S(\mathcal{M},\tau)\) with respect to the measure topology forms the \(\tau\)-compact operators.
In this setting, the authors consider the sets \(I_B=\{A\in S(\mathcal{M},\tau): A=A^*$, $-B\le A\le B\}\) and \(K_B=\{A\in S(\mathcal{M},\tau): A^*A\le B\}\) for positive \(B\in S(\mathcal{M},\tau)\).
They show that both \(I_B\) and \(K_B\) are convex and closed in measure and can be written in the form \(I_B=\{\sqrt{B}T\sqrt{B}: T\in\mathcal{M},\,T=T^*, \,\|T\|_\infty\leq1\}\) and \(K_B=\{T\sqrt{B}: T\in\mathcal{M},\, \|T\|_\infty\leq1\}\). The authors show that, if \(\mathcal{M}\) is atomic, then \(I_B\) is \(t_\tau\)-compact if and only if \(B\) is and that this characterizes atomic von Neumann algebras \((\mathcal{M},\tau)\); under a mild additional hypothesis, a Krein-Milman theorem with respect to \(t_\tau\) holds for \(I_B\). If \(I_B\) is considered in certain symmetrically normed operator spaces \(E(\mathcal{M},\tau)\) (which include the predual \(S_1(H)\) of \(B(H)\)), then a sequence in \(I_B\), converging locally in measure, converges in norm. To put this in perspective, note that, as the authors point out, in general in \(S_1(H))\), a sequence converging weakly and locally in measure need not converge in norm, contrary to what happens in the commutative case.

46L51 Noncommutative measure and integration
46L52 Noncommutative function spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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