A Non-commutative Wiener-Wintner Theorem

For a von Neumann algebra $\cal M$ with a faithful normal tracial state $\tau$ and a positive ergodic homomorphism $\alpha:\mathcal L^1(\mathcal M,\tau)\to \mathcal L^1(\mathcal M,\tau)$ such that $\alpha$ does not increase the norm in $\mathcal M$ and $\tau \circ \alpha=\tau$, we establish a non-commutative counterpart of the classical Wiener-Wintner Theorem.