Weak sequential completeness of factor spaces of the space of integrable operators
An essential ingredient for the characterization of the subspaces \(X\) of \(L^1(S,\Sigma,\mu)\), which are \(L\)-summands in their biduals, are results of Bukhvalov and Lozanovskii on sets closed with respect to convergence in measure [\textit{A. V. Bukhvalov} and \textit{G. Ya. Lozanovskii}, Trans. Mosc. Math. Soc. 34, 127--148 (1978; Zbl 0425.46019); see also \textit{P. Harmand, D. Werner} and \textit{W. Werner}, ``\(M\)-ideals in Banach spaces and Banach algebras.'' Lecture Notes in Mathematics 1547. Berlin: Springer-Verlag (1993; Zbl 0789.46011)]. For those \(X\) one obtains, that \(L^1(\mu)/X\) is weakly sequentially complete. The work of Bukhvalov and Lozanovskij was partly extended to the noncommutative case by \textit{G. Sh. Skvortsova} and \textit{O. E. Tikhonov} [Russ. Math. 42, 46--52 (1999; Zbl 1115.46306)]. More precisely, a semifinite von Neumann algebra \(M\), a faithful, normal, semifinite trace \(\tau\) on \(M\), and the space \(L^1(\tau)\) are considered. These results are applied in the present paper to subspaces \(X\) of \(L^1(\tau)\). It is shown that the unit ball of \(X\) is closed with respect to convergence in measure, and hence \(L^1(\tau)/X\) is weakly sequentially complete, if \(X\) is one of the following: (1) a certain space constructed from a family of projections in \(M\), (2) a noncommutative Hardy space build from a suitable group of \(*\)-automorphisms, (3) a space of integrable operators constructed for certain subalgebras \(N\) of \(M\).
scientific article; zbMATH DE number 1960504