Ergodic theorems for free group actions on von Neumann algebras
The present paper extends a recent ergodic theorem of \textit{A. Nevo} and \textit{E. M. Stein} [Acta Math. 173, 135-154 (1994; Zbl 0837.22008)] to the noncommutative case. Let \(\{a_i\}\), \(i=1, \dots,r\), generate the free group \(F_r\) on \(r\) generators. Let \(\rho\) denote a faithful normal state on a von Neumann algebra \(A\), and let \(\{\alpha_i\}\), \(i=1, \dots,r\), be *-automorphisms of \(A\) leaving \(\rho\) invariant. Define the group homomorphism \(\varphi\) from \(F_r\) to *-automorphisms of \(A\) by \(\varphi(a_i) =\alpha_i\). Let \(\omega_n\) denote the set of all reduced words in \(F_r\) of length \(n\). Set \[ \sigma_n ={1\over |\omega_n|} \sum_{a\in \omega_n} \varphi(a), \quad S_n= {1\over n} \sum^{n-1}_{k=0} \sigma_k. \] The main theorem states that \(S_n(x)\) converges almost uniformly for \(x\) in \(A\).
scientific article; zbMATH DE number 1080228