Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their \(L_{2}\)-spaces (Q2502998)

Let \({\mathfrak A}\) be a von Neumann algebra and \(\varphi\) be a faithful normal state on \({\mathfrak A}\). Operators \(A,B\in{\mathfrak A}\) are said to be uncorrelated if \(\text{Cov}(A,B):= \varphi(B^*A)- \varphi(A)\varphi(B^*)\) vanishes. The main result asserts that if a sequence \((X_n)_{n\geq 1}\) of uncorrelated operators in \({\mathfrak A}\) is bundle convergent to an operator \(X\in{\mathfrak A}\) and \(\sum^\infty_{n=1} n^{-2}\text{Var}(X_n)\cdot\log^2(n+1)< \infty\), then \(X\) is a scalar multiple of identity. Here, \(\text{Var}(X_n)= \text{Cor}(X_n, X_n)\), \(n\geq 1\), and bundle convergence is a noncommutative generalization of the classical a.\,s.\ convergence introduced by \textit{E.~Hensz}, \textit{R.~Jajte} and \textit{A.~Paszkiewicx} [Stud.\ Math.\ 120, No.~1, 23--46 (1996; Zbl 0856.46033)].