  In the paper we consider $T_{1},..., T_{d}$ absolute contractions of von
Neumann algebra $\M$ with normal, semi-finite, faithful trace, and prove that
for every bounded Besicovitch weight $\{a(\kb)\}_{\kb\in\bn^d}$ and every $x\in
L_{p}(\M)$, ($p>1$) the averages
  A_{\Nb}(x)=\frac{1}{|\Nb|}\sum\limits_{\kb=1}^{\Nb}a(\kb)\Tb^{\kb}(x).
converge bilaterally almost uniformly in $L_{p}(\M)$.
