Invariant mean - Imported Knowlpedia Content

Definition

Let $G$ be a topological group and let $\mathcal A$ be a specified unital, self-adjoint, translation-invariant space of bounded complex functions on $G$ . A left-invariant mean on $\mathcal A$ is a linear functional

m:\mathcal A\longrightarrow\mathbb C

such that $m(f)\geq0$ whenever $f\geq0$ , $m(1)=1$ , and

m(L_gf)=m(f),\qquad (L_gf)(x)=f(g^{-1}x),

for all $g\in G$ . Thus a mean is a normalized positive averaging functional, while invariance says that translating the input does not change its average. The domain is part of the notion: standard choices include $\operatorname{RUC}_b(G)$ , the bounded right-uniformly continuous functions, and, for locally compact $G$ , $L^\infty(G)$ modulo null sets.

Positivity and norm

Positivity and $m(1)=1$ imply $\lVert m\rVert=1$ , so a mean is automatically continuous in the supremum norm. It also satisfies $\inf f\leq m(f)\leq\sup f$ for real-valued $f$ , and $m(\overline f)=\overline{m(f)}$ . Unlike evaluation at a point, an invariant mean need not be countably additive or represented by a probability measure on $G$ .

Amenability and examples

A standard definition calls $G$ amenable when such a mean exists on $\operatorname{RUC}_b(G)$ ; for locally compact $G$ , this is equivalent to existence on $L^\infty(G)$ . For a compact group, integration against normalized Haar measure gives an invariant mean. Locally compact abelian groups and solvable locally compact groups are amenable, whereas the discrete free group on two generators is not. These examples and the equivalence with other locally compact formulations are treated in Paterson, Chapters 0–1.

Conventions and function spaces

For a discrete group, $\operatorname{RUC}_b(G)=\ell^\infty(G)$ . For a locally compact group, amenability is also equivalent to the existence of an invariant mean on $L^\infty(G)$ , with translations understood modulo null sets. Authors may instead use left-uniformly continuous functions and right translations.

References

A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, 1988. AMS DOI record. Relevant: Chapters 0–1 on invariant means and amenable groups.