Bundle convergence of harmonic means of orthogonal sequences in noncommutative \(L_2\)-spaces (Q1602603)

Let \(A\) be a von Neumann algebra, \(\varphi\) a faithful normal state, \(L_2= L_2(A,\varphi)\). Let \(\xi_k@> b>> 0\) mean the bundle convergence of \((\xi_k)\subset L_2\), to zero \(0\in L_2\). The main results of the paper: Theorem 1. If \((\xi_k)\) is an orthogonal sequence in \(L_2\) and \(\sum^\infty_{k=1} {\|\xi_k\|^2\over k^2}< \infty\) then \(\tau_n@>b>> 0\) as \(n\to\infty\). (Here \(\tau_n= (\sum^n_{k=1} {\xi_k\over k})/(\sum^n_{k=1} {1\over k})\) is the harmonic mean of first-order of \((\xi_k)\).) Theorem 2. If \((\xi_k)\) is an orthogonal sequence in \(L_2\) and \(\sum^\infty_{k=2} {\|\xi_k\|^2\over k^2(\ln\ln k)^2}< \infty\), then \(\tau^{(2)}_n@>b>> 0\) as \(n\to\infty\). (Here \(\tau^{(2)}_n\) is the harmonic mean of second-order.) The last result seems to be new even in the commutative case.