Strong limit theorems for orthogonal sequences in von Neumann algebras

Abstract:

Let $A$ be a von Neumann algebra with a faithful normal state $\phi$. It is shown that if a sequence $({x_n})$ in $A$ is orthogonal relative to $\phi$ and satisfies the condition \[ \sum \limits _{k} {\phi (|{x_k}{|^2}){{\left ( {\frac {{\log k}}{k}} \right )}^2} < \infty ,} \] then ${}_n^1\sum \nolimits _{k = 1}^n {{x_k} \to 0}$ almost uniformly in $A$. Some other results related to this theorem are also discussed.