On mixing and completely mixing properties of positive L ¹-contractions of finite von Neumann algebras

Akcoglu and Suchaston proved the following result: Let T : L ¹(X,F,μ) → L¹(X, F, μ) be a positive contraction. Assume that for z ∈ L¹(X,F,μ) the sequence (Tⁿz) converges weakly in L¹(X,F,μ). Then either lim_n→∞ ∥Tⁿz∥ = 0 or there exists a positive function h ∈ L¹(X, F,μ),h ≠ 0 such that Th = h. In the paper we prove an extension of this result in a finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative L¹-space has no nonzero positive invariant element, then its mixing property implies the completely mixing property.