On noncommutative weighted local ergodic theorems on L ^p -spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ε L ^p (M, τ), that the averages 1/T ∫ ₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0.