ALT Let ℳ be a semifinite von Neumann algebra and T: ℳ → ℳ be a positive L_∞-L₁ contraction in the sense of Junge-Xu, of which the numerical range, as an operator on L₂(ℳ), is contained in a closed polygon with vertices on the unit circle. Let 1<p<∞. In this article, we prove that there exists a positive constant C_p(T) such that \Big‖sup_n≥0⁺ T^n x \Big‖_p ≤ C_p(T) ‖x‖_p for all x ∈ L_p(ℳ), extending some noncommutative maximal ergodic inequalities proved by Junge-Xu and later generalized by Bekjan. In the commutative setting, the similar inequalities as above hold for arbitrary L_∞-L₁ contractions with the same condition on the numerical range, yielding a vast generalisation of a classical maximal ergodic theorem of Stein proved in 1960s. We further prove a variational inequality for contractively regular operators T:L_p(Ω)→ L_p(Ω) whose peripheral spectrum is finite and satisfies a suitable resolvent estimate, extending earlier work of Le Merdy and Xu. Finally, we establish a noncommuta