Ergodic Theorems for Free Group Actions on von Neumann Algebras

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 {ai}ri=1generateFr, the free group onrgenerators, and let {αi}ri=1be *-automorphisms ofAwhich leaveρinvariant. Defineφto be the group homomorphism fromFrto the *-automorphisms ofAdefined on base elements byφ: ai↦αi. Definewnas the set of all reduced words inFrof lengthn(the identity is a neutral element), and |wn| as the number of elements ofwn. Letσn=(1/|wn|) ∑a∈wn φ(a) andSn=(1/n) ∑n−1k=0 σk. We then show that ifxis inA, thenSn(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,σnpositive maps such thatσ1∘σn=p1σn+1+p2σn−1(n⩾1), withσ0(x)=xandσ1(1)=1 then, with a few technical assumptions, we show a convergence result for limn→∞ σn(x) and show that limn→∞ n−1∑n−1k=0 σk(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 Guivarcʹh