NONCOMMUTATIVE VERSION OF KOLMOGOROV'S THREE SERIES THEOREM AND SOME LIMIT THEOREMS

In this note a noncommutative version of Kolmogorov's Three Series Theorem, concerning unconditional convergence, is given. As a noncommutative counterpart of the classical almost sure convergence the almost uniform convergence of measurable operators is used. We also provide a method of proving theorems on almost uniform convergence in von Neumann algebras. Finally, we give a noncommutative version of the Kolmogorov's 0-1 law and show that (under appropriate assumptions concerning independence) the limit of a convergent sequence of elements of a von Neumann algebra is a multiplicity of identity.