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Harmonic analysis and pointwise ergodic theorems for noncommuting transformations

Author: Amos Nevo
Journal: J. Amer. Math. Soc. 7 (1994), 875-902
MSC: Primary 22D40; Secondary 28D15, 43A80
MathSciNet review: 1266737
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Abstract: Let $ {F_k}$ denote the free group on $ k$ generators, $ 1 < k < \infty $, and let $ S$ denote a set of free generators and their inverses. Define $ {\sigma _n} \stackrel{d}{=} \frac{1}{{\char93 {S_n}}}{\Sigma _{w \in {S_n}}}w$, where $ {S_n} = \{ w:\vert w\vert = n\} $, and $ \vert \cdot \vert$ denotes the word length on $ {F_k}$ induced by $ S$. Let $ (X, \mathcal{B}, m)$ be a probability space on which $ {F_k}$ acts ergodically by measure preserving transformations. We prove a pointwise ergodic theorem for the sequence of operators $ \sigma _n^\prime = \frac{1}{2}({\sigma _n} + {\sigma _{n + 1}})$ acting on $ {L^2}(X)$, namely: $ \sigma _n^\prime f(x) \to \int_X {f\,dm} $ almost everywhere, for each $ f$ in $ {L^2}(X)$. We also show that the sequence $ {\sigma _{2n}}$ converges to a conditional expectation operator with respect to a $ \sigma $-algebra which is invariant under $ {F_k}$. The proof is based on the spectral theory of the (commutative) convolution subalgebra of $ {\ell ^1}({F_k})$ generated by the elements $ {\sigma _n},\,\;n \geq 0$. We then generalize the discussion to algebras arising as a Gelfand pair associated with the group of automorphisms $ G({r_1},\;{r_2})$ of a semi-homogeneous tree $ T({r_1},\;{r_2})$, where $ {r_1} \geq 2,\;{r_2} \geq 2,\;{r_1} + {r_2} > 4$. (The case of $ {F_k}$ corresponds to that of a homogeneous tree of valency $ 2k$.) We prove similar pointwise ergodic theorems for two classes of subgroups of $ G({r_1},\;{r_2})$. One is the class of closed noncompact boundary-transitive subgroups, including any simple algebraic group of split rank one over a local field, for example, $ PS{L_2}({\mathbb{Q}_p})$. The second class is that of lattices complementing a maximal compact subgroup. We also prove a strong maximal inequality in $ {L^2}(X)$ for the groups listed above, as well as a mean ergodic theorem for unitary representations of the groups (due to $ {\text{Y}}$. Guivarc'h for $ {F_k}$). Finally, we describe the structure and spectral theory of a noncommutative algebra which arises naturally in the present context, namely the double coset algebra associated with the subgroup of $ G({r_1},\;{r_2})$ stabilizing a geometric edge. The results are applied to prove mean ergodic theorems for a family of lattices in $ G({r_1},\;{r_2})$, which includes, for example, $ PS{L_2}(\mathbb{Z})$.

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Keywords: Harmonic analysis, pointwise ergodic theorems, semi-homogeneous trees, free groups, convolution algebras, Gelfand pairs, maximal inequality
Article copyright: © Copyright 1994 American Mathematical Society