# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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## Harmonic analysis and pointwise ergodic theorems for noncommuting transformationsHTML articles powered by AMS MathViewer

by Amos Nevo
J. Amer. Math. Soc. 7 (1994), 875-902 Request permission

## 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 ${\sigma _n} \stackrel {d}{=} \frac {1}{{\# {S_n}}}{\Sigma _{w \in {S_n}}}w$, where ${S_n} = \{ w:|w| = n\}$, and $| \cdot |$ 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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