A non-commutative Wiener–Wintner theorem

For a von Neumann algebra M with a faithful normal tracial state τ and a positive ergodic homomorpsism α: L¹ (M, τ) → L¹ (M, τ) such that τ ◦ α = τ and α does not increase the norm in M, we establish a non-commutative counterpart of the classical Wiener–Wintner ergodic theorem.