BESICOVICH FUNCTIONS AND WEIGHTED ERGODIC THEOREMS FOR LCA GROUP ACTIONS

WILLIAM CASPER

Abstract. This is an abstract.

1. Introduction

Throughout this paper, we define G to be a locally compact group with (left) Haar measure λ(If G is compact, we assume the Haar measure is normalized, so that λ(G) = 1.). We denote by C(G) the collection of all continuous, complex-valued functions on G, and by C_{00}(G) the collection of all continuous, complex-valued functions on G with support contained in a compact set. Additionally, we denote by C_{0}(G) the collection of all continuous, complex-valued functions on G which become arbitrarily small outside a compact set, i.e. given ϵ > 0, there exists a compact set C ⊂ G such that |f(x)| < ϵ for all x ∉ C. For any linear space F of complex-valued functions on G, we denote by F^{r} and F^{+} the collection of all real-valued functions and positive functions in F, respectively. Additionally, we use F^{∗} to denote the dual space of F. We denote by M(G) the collection of all complex measures on G.

1.1. Basic Definitions and Facts.

Definition 1. Let F be a linear space of complex-valued functions on G such that for every f ∈ F, the function x ↦ f(a^{−1}x) (x ↦ f(xa^{−1})) is in F for all a ∈ G. Then F is called left translation invariant (right translation invariant). For any fixed a ∈ G, we define the operators

L(a) : f ↦ L(a)f, where L(a)f : x ↦ f(a^{−1}x);
R(a) : f ↦ R(a)f, where R(a)f : x ↦ f(xa^{−1}).

Given a ∈ G, the function L(a) : f ↦ L(a)f (R(a) : f ↦ R(a)f) are linear operators on any left (right) translation invariant space F. The left regular representation L and right regular representation R on G are defined respectively by

L : a ↦ L(a), where L(a) : f ↦ L(a)f;
R : a ↦ R(a), where R(a) : f ↦ R(a)f.

Fact 2. The left and right regular representations are representations of G. The spaces C(G), C_{00}(G), and C_{0}(G) are left and right translation invariant.

Date: February 15, 2010.

Definition 3. Let μ ∈ M(G) and f ∈ L_{p}(G) for 1 ≤ p ≤ ∞. We define the convolution of μ and f by

μ ∗ f(x) = ∫_{G} f(y^{−1}x)dμ(y).

The convolution of two measures μ, ν ∈ M(G), is the measure μ ∗ ν defined by

μ ∗ ν(E) = ∫_{G} ∫_{G} 1_{E}(xy)dν(y)dμ(x)

for all Borel sets E.

Fact 4. If μ ∈ M(G) and f ∈ L_{p}(G) for 1 ≤ p ≤ ∞, then μ ∗ f ∈ L_{p}. For any a ∈ G, δ_{a} ∗ f = L(a)f. If dν = fdλ, then using Fubini’s theorem and the left translation invariance of λ,

μ ∗ ν(E) = ∫_{G} ∫_{G} 1_{E}(xy)f(y)dydμ(x) = ∫_{G} ∫_{G} 1_{E}(y)f(x^{−1}y)dydμ(x)
= ∫_{E} ∫_{G} f(x^{−1}y)dμ(x)dy = ∫_{E} μ ∗ f(y)dy.

Thus d(μ ∗ ν) = μ ∗ fdλ.

Definition 5. A bounded, continuous function f is (weakly) almost periodic if the set {L(a)f : a ∈ G} is (weakly) conditionally compact in C(G). The collection of all (weakly) almost periodic functions is denoted by (WAP(G)) AP(G).

Definition 6. A sequence {μ_{n}}_{n=1}^{∞} converges weakly to invariance if for every f ∈ C(G) and t ∈ G, we have

(1)  ∫_{G} f(x^{−1}) − f(x^{−1}t)dμ_{n}(x) = (μ_{n} ∗ f)(e) − (μ_{n} ∗ L(t)f)(e) = 0.

A sequence is ergodic if for every f ∈ L_{1}(G), and every t ∈ G, we have

(2)  lim_{n→∞} ∥μ_{n} ∗ (f − L(t)f)∥_{1} = 0.

Example 7. Let μ ∈ M(G) and define {μ_{k}}_{k=1}^{∞} ⊂ M(G) by μ_{1} = μ and μ_{k+1} = μ ∗ μ_{k} for k ≥ 1. Let f ∈ L_{2}(G). In general, the sequence of functions {μ_{k} ∗ f}_{k=1}^{∞} does not converge either pointwise or in L_{2}(G). As an example, let G be a finite group and let S be a symmetric set of generators for G. The Haar measure λ on G is the normalized counting measure. Define a measure μ(A) = λ(A ∩ S)/λ(A) for all subsets A ⊂ G. Then μ is a probability measure on G. If e ∈ S, then

μ_{k} ∗ f → \frac{1}{|G|}

in L_{2}(G) for all functions f : G → ℂ. To see this, let f : G → C and let Ĝ represent the dual of G. For every equivalence class of representations σ ∈ G, fix a representation π_{σ} ∈ σ with corresponding representation space H_{σ} of dimension d_{σ} and let {ξ_{i}^{(σ)}}_{i=1}^{d_{σ}} be a fixed basis of H_{σ}. Define π_{ij}^{(σ)} to be the coordinate map π_{ij}^{(σ)}(x) = ⟨π_{σ}(x)ξ_{i}^{(σ)}, ξ_{j}^{(σ)}⟩. Then f ∈ L_{2}(G) and Peter-Weyl theorem tells us that

f(x) = ∑_{σ∈Ĝ} ∑_{i,j=1}^{d_{σ}} f_{ij}^{(σ)}\sqrt{d_{σ}}π_{ij}^{(σ)}(x)

for some complex numbers f_{ij}^{(σ)} ∈ ℂ satisfying

∥f∥_{2} = (∑_{σ∈Ĝ} ∑_{i,j=1}^{d_{σ}} |f_{ij}^{(σ)}|^{2})^{2} .

Since

δ_{a} ∗ π_{ij}^{(σ)}(x) = π_{ij}^{(σ)}(a^{−1}x) = ∑_{k=1}^{d_{σ}} π_{ik}^{(σ)}(a^{−1})π_{kj}^{(σ)}(x)

It follows that

(μ ∗ f)(x) − ∫_{G} fdλ = ∑_{σ∈Ĝ} ∑_{i,j=1}^{d_{σ}} f_{ij}^{(σ)}\sqrt{d_{σ}} \frac{1}{|S|} ∑_{s∈S \\{e\}} ∑_{k=1}^{d_{σ}} π_{ik}^{(σ)}(s^{−1})π_{kj}^{(σ)}(x)

+ ∑_{σ∈Ĝ} ∑_{i,j=1}^{d_{σ}} f_{ij}^{(σ)}\sqrt{d_{σ}} \frac{1}{|S|} (π_{ij}^{(σ)}(x) − \frac{|S|}{|G|} ∑_{y∈G} π_{ij}^{(σ)}(y)) .

Therefore

∥(μ ∗ f)(x) − ∫_{G} fdλ∥_{2} ≤ (\frac{|S| − 1}{|S|}) ∥f∥_{2} + (\frac{|G| − 1}{|G|}) ∥f∥_{2} + (\frac{1}{|S|} − \frac{1}{|G|}) ∥f∥_{2}

In particular, it follows that the sequence{μ_{k}}_{k=1}^{∞} is ergodic and converges weakly to invariance.

If e ∉ S, then this is not necessarily the case. In particular, if G = S_{6} and S consists of all two-cycles of G, then μ_{k} ∗ 1_{{e}} does not converge. Instead, it cycles through a finite number of values. I do think it turns out to be ergodic, however. Details still need to be worked out.

2. Almost Periodic Functions

3. Convolution as a Markov Operator

Appendix A. Examples

Appendix B. Proofs of Theorems

References