Harmonic analysis and pointwise ergodic theorems for noncommuting transformations

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, $PS{L_2}(\mathbb {Z})$.