Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators
From the authors' abstract: Given a von Neumann algebra \(M\) with a faithful normal semifinite trace \({\tau}\), the authors prove that each order bounded sequence of \({\tau}\)-compact operators has a subsequence whose arithmetic averages converge in \({\tau}\). They also prove a noncommutative analog of Pratt's lemma for \(L_{1}(M, {\tau})\). They apply the main result to \(L_{p}(M, {\tau})\) with \(0<p \leq 1\) and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of \({\tau}\)-compactness of the dominant.
scientific article; zbMATH DE number 6105672
scientific article