Ergodic Theorems for Free Group Actions on von Neumann Algebras
✍ Scribed by Trent E. Walker
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 344 KB
- Volume
- 150
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
We extend a recent ergodic theorem of A. Nevo and E. Stein to the non-commutative case. Let \ be a faithful normal state on the von Neumann algebra A.
Let [a i ] r i=1 generate F r , the free group on r generators, and let [: i ] r i=1 be V-automorphisms of A which leave \ invariant. Define , to be the group homomorphism from F r to the *-automorphisms of A defined on base elements by ,: a i [ : i . Define w n as the set of all reduced words in F r of length n (the identity is a neutral element), and |w n | as the number of elements of w n . Let _ n =(1Â|w n |) a # w n ,(a) and S n =(1Ân) n&1 k=0 _ k . We then show that if x is in A, then S n (x) converges almost uniformly to an element x^# 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, if p 1 p 2 0, p 1 +p 2 =1, _ n positive maps such that _ 1 b _ n =p 1 _ n+1 +p 2 _ n&1 (n 1), with _ 0 (x)=x and _ 1 (1)=1 then, with a few technical assumptions, we show a convergence result for lim n Ä _ n (x) and show that lim n Ä n &1 n&1 k=0 _ k (x) converges almost uniformly. In the case p 2 =0, _ 1 a *-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.