Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circ