Harmonic analysis and pointwise ergodic theorems for noncommuting transformations

Nevo, Amos

Journal of the American Mathematical Society, Tome 07 (1994) no. 4, pp. 875-902

Résumé

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

Détail

DOI : 10.1090/S0894-0347-1994-1266737-5

Bibliographie

[1] Bouaziz-Kellil, Ferdaous Représentations sphériques des groupes agissant transitivement sur un arbre semi-homogène Bull. Soc. Math. France 1988 255 278

[2] Cartier, P. Harmonic analysis on trees 1973 419 424

[3] Dunford, Nelson, Schwartz, Jacob T. Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space 1963

[4] Eskin, Alex, Mcmullen, Curt Mixing, counting, and equidistribution in Lie groups Duke Math. J. 1993 181 209

[5] Furstenberg, Hillel, Katznelson, Yitzchak, Weiss, Benjamin Ergodic theory and configurations in sets of positive density 1990 184 198

[6] Figã -Talamanca, Alessandro, Nebbia, Claudio Harmonic analysis and representation theory for groups acting on homogeneous trees 1991

[7] Figã -Talamanca, Alessandro, Picardello, Massimo A. Harmonic analysis on free groups 1983

[8] Guivarc’H, Yves Généralisation d’un théorème de von Neumann C. R. Acad. Sci. Paris Sér. A-B 1969

[9] Howe, Roger E., Moore, Calvin C. Asymptotic properties of unitary representations J. Functional Analysis 1979 72 96

[10] Iozzi, Alessandra Harmonic analysis on the free product of two cyclic groups Boll. Un. Mat. Ital. B (6) 1985 167 177

[11] Iozzi, Alessandra, Picardello, Massimo A. Spherical functions on symmetric graphs 1983 344 386

[12] Iwahori, N., Matsumoto, H. On some Bruhat decomposition and the structure of the Hecke rings of 𝔭-adic Chevalley groups Inst. Hautes Études Sci. Publ. Math. 1965 5 48

[13] Jones, Roger L. Ergodic averages on spheres J. Anal. Math. 1993 29 45

[14] Jones, Roger, Rosenblatt, Joseph, Tempelman, Arkady Ergodic theorems for convolutions of a measure on a group Illinois J. Math. 1994 521 553

[15] Matsumoto, Hideya Analyse harmonique dans les systèmes de Tits bornologiques de type affine 1977

[16] Macdonald, I. G. Spherical functions on a group of 𝑝-adic type 1971

[17] Nevo, Amos A structure theorem for boundary-transitive graphs with infinitely many ends Israel J. Math. 1991 1 19

[18] Nevo, Amos, Stein, Elias M. A generalization of Birkhoff’s pointwise ergodic theorem Acta Math. 1994 135 154

[19] Oseledec, V. I. Markov chains, skew products and ergodic theorems for “general” dynamic systems Teor. Verojatnost. i Primenen. 1965 551 557

[20] Satake, Ichir㴠Theory of spherical functions on reductive algebraic groups over 𝔭-adic fields Inst. Hautes Études Sci. Publ. Math. 1963 5 69

[21] Stein, E. M. On the maximal ergodic theorem Proc. Nat. Acad. Sci. U.S.A. 1961 1894 1897

[22] Stein, Elias M. Topics in harmonic analysis related to the Littlewood-Paley theory. 1970

[23] Stein, Elias M. Maximal functions. I. Spherical means Proc. Nat. Acad. Sci. U.S.A. 1976 2174 2175

[24] Weiss, Benjamin Positive cones in Hilbert space and a maximal inequality 1972 353 358