Ergodic Theorems for Free Group Actions on von Neumann Algebras

Abstract

We extend a recent ergodic theorem of A. Nevo and E. Stein to the non-commutative case. Letρbe a faithful normal state on the von Neumann algebraA. Let {a_i}^r_i=1generateF_r, the free group onrgenerators, and let {α_i}^r_i=1be *-automorphisms ofAwhich leaveρinvariant. Defineφto be the group homomorphism fromF_rto the *-automorphisms ofAdefined on base elements byφ: a_i↦α_i. Definew_nas the set of all reduced words inF_rof lengthn(the identity is a neutral element), and |w_n| as the number of elements ofw_n. Letσ_n=(1/|w_n|) ∑_a∈wn φ(a) andS_n=(1/n) ∑ⁿ⁻¹_k=0 σ_k. We then show that ifxis inA, thenS_n(x) converges almost uniformly to an element○∈A. To prove the above theorem, we prove an ergodic theorem involving completely positive maps, of which the free group situation is a special case. Roughly, ifp₁⩾p₂⩾0,p₁+p₂=1,σ_npositive maps such thatσ₁∘σ_n=p₁σ_n+1+p₂σ_n−1(n⩾1), withσ₀(x)=xandσ₁(1)=1 then, with a few technical assumptions, we show a convergence result for lim_n→∞ σ_n(x) and show that lim_n→∞ n⁻¹∑ⁿ⁻¹_k=0 σ_k(x) converges almost uniformly. In the casep₂=0,σ₁a *-automorphism, our theorems correspond to the non-commutative pointwise ergodic theorem of E. C. Lance. The results partially generalize a result of Kummerer. Our theorems also include results concerning normal operators on a Hilbert space which generalizes work of Guivarc'h