Amenable locally compact group - Imported Knowlpedia Content

Definition

A locally compact group $G$ is amenable if there exists a left invariant mean on $L^\infty(G)$ . Concretely, this is a linear functional

M:L^\infty(G)\longrightarrow\mathbb C

that is positive, satisfies $M(1)=1$ , and obeys

M(L_gf)=M(f) \qquad\text{for all }g\in G,

where $L_gf(x)=f(g^{-1}x)$ . Here $L^\infty(G)$ is formed using any left Haar measure; its measure class is independent of the normalization. Amenability says that invariant averaging exists even when Haar measure cannot be normalized to have finite total mass.

Equivalent viewpoints

Reiter’s condition $P_1$ is equivalent to amenability: there is a net of nonnegative functions $\varphi_i\in L^1(G)$ with $\lVert\varphi_i\rVert_1=1$ such that

\lVert L_g\varphi_i-\varphi_i\rVert_1\longrightarrow 0

uniformly for $g$ in compact subsets of $G$ . Another equivalent condition is the fixed-point property: every continuous affine action of $G$ on a compact convex subset of a locally convex space has a fixed point. These forms connect averaging, approximation, and dynamics.

Examples and permanence

Compact groups are amenable because normalized Haar integration is an invariant mean. Locally compact abelian groups and solvable locally compact groups are amenable. Closed subgroups, quotients, extensions, and directed unions preserve amenability under the standard locally compact hypotheses. The discrete free group on two generators is the basic non-amenable example.

Operator-algebraic significance

Amenability can also be detected by weak containment of the trivial representation in the regular representation. For group operator algebras, it is equivalent to injectivity of the canonical map from the full to the reduced group $C^*$ -algebra; this equivalence is isolated in amenability and equality of full and reduced group $C^*$ -algebras.

The invariant-mean and Reiter formulations are developed in Paterson, Amenability.

References

Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, 1988. AMS DOI record. Relevant: invariant means and amenability of locally compact groups.
Frederick P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand Mathematical Studies 16, 1969. Google Books record. Relevant: invariant means on locally compact groups and their applications.