On noncommutative weighted local ergodic theorems

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In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace, and {α t} a strongly continuous extension to Lᵖ(M,) of a semigroup of absolute contractions on L¹ (M,τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x∈ Lᵖ(M,), the averages 1/T∫₀ᵗb(t)ₜ(x)dt converge bilateral almost uniform in Lᵖ(M,) as T→ 0.