Equicontinuity in measure spaces and von Neumann algebras
Let \(X\) be a Banach space and denote by \(cabv(\Sigma,X)\) the countably additive \(X\)-valued measures of bounded variation on \(\Sigma\). A sequence \((F_n)\subset cabv(\Sigma,X)\), absolutely continuous with respect to some probability measure \(\mu\), is said to converge \(w^2\) to \(F\) if (1) there exists a sequence of measurable sets \(B_k\searrow B\) and \(\mu(B)=0\) and (2) for each \(k\), if there is a measurable \(A\subset B_k^c\), then \(F_n(A)\) converges weakly to \(F(A)\) in \(X\). The author's Biting Lemma says: When \(X\) is the scalar field, every bounded sequence in \(cabv(\Sigma,X)\) has a \(w^2\)-convergent subsequence. The author now takes \(X\) reflexive and \((F_n)\subset cabv(\Sigma,X)\) such that \(F_n\ll\mu\) and proves that \((F_n)\) has a subsequence that converges \(w^2\) to some \(F\) in \(cabv(\Sigma,X)\). Moreover, \(F\ll\mu\). The proof uses the scalar Biting Lemma. The author informs in a remark that the results also holds if \(X\) and \(X^\ast\) enjoy the RNP. The \(C^\ast\)-algebra version of the Biting Lemma was established in [\textit{K.\,K.\,Brooks, K.\,Saitô} and \textit{J.\,D.\,M.\,Wright}, J.~Math.\ Anal.\ Appl.\ 276, No.\,1, 160--167 (2002; Zbl 1018.46031)]. Two theorems concerning weakly compact operators on some \(C^\ast\)-algebras are announced at the end of the paper.
scientific article; zbMATH DE number 5070777
scientific article