  In the present paper we consider a von Neumann algebra M with a faithful
normal semi-finite trace $\t$, and $\{\alpha_ t\} $ a strongly continuous
extension to $L^p(M,\t)$ of a semigroup of absolute contractions on $L^1
(M,\tau)$. By means of a non-commutative Banach Principle we prove for a
Besicovitch function b and $x\in L^p(M,\t)$, the averages
\frac{1}{T}\int_0^Tb(t)\a_t(x)dt converge bilateral almost uniform in
$L^p(M,\t)$ as $T\to 0$.
