Noncommutative extensions of an ergodic theorem of Breiman

Let (M,τ) be a tracial von Neumann algebra, T:L₁(M,τ)→ L₁(M,τ) be induced by a normal $τ$-preserving $*$-automorphism of M, and x∈ L₁(M,τ). We will prove the bilaterally almost uniform convergence of averages of the form 1/n∑ₖ₌₀ⁿ⁻¹Tᵏ(xₖ), generalizing an ergodic theorem proven by Breiman in {Breiman57}, where (xₖ)ₖ₌₀∞ is an element of the noncommutative vector-valued Lₚ-space Lₚ(M,τ;_∞) such that there exists x∈ Lₚ(M,τ) with (xₖ-x)ₖ₌₀∞∈ Lₚ(M,τ;c₀). A generalization to semifinite (M,τ) and general positive Dunford-Schwartz operators T∈ DS⁺(M,τ) is shown as well when $1