Amenability via weak containment of the trivial representation

Definition

Let $G$ be a locally compact group, let $1_G$ denote its one-dimensional trivial unitary representation, and let $\lambda_G$ be its left regular representation on $L^2(G)$ . The Hulanicki–Reiter criterion states that

G\text{ is amenable}\quad\Longleftrightarrow\quad 1_G\prec\lambda_G,

where $\prec$ denotes weak containment. Equivalently, there is a net of unit vectors $\xi_i\in L^2(G)$ such that

\sup_{g\in C}\lVert\lambda_G(g)\xi_i-\xi_i\rVert_2\longrightarrow0

for every compact $C\subseteq G$ . Thus amenability is detected by almost invariant vectors in the regular representation.

Why the formulations agree

The almost-invariant-vector condition says that the constant coefficient $1$ of $1_G$ is uniformly approximated on compact sets by positive-definite coefficients of $\lambda_G$ , which is precisely weak containment for the trivial representation. Squaring absolute values, $\lvert\xi_i\rvert^2$ , turns such vectors into approximately invariant $L^1$ -probability densities; conversely, taking square roots of Reiter $P_1$ densities yields almost invariant $L^2$ -vectors. This is the Hulanicki–Reiter bridge between representation theory and invariant-mean amenability Bekka–de la Harpe–Valette, §G.3.

Operator-algebraic consequence

Weak containment $\pi\prec\sigma$ is equivalent to the associated integrated representations satisfying $\ker\sigma\subseteq\ker\pi$ on the full group $C^*$ -algebra. Applied to $1_G$ and $\lambda_G$ , the criterion identifies amenability with continuity of the trivial representation for the reduced group- $C^*$ -norm. This is one route to the equality of full and reduced group $C^*$ -algebras for amenable groups.

Examples and scope

For an abelian locally compact group, normalized functions supported on increasingly invariant sets provide the required almost invariant vectors. For the free group on two generators, the trivial representation is not weakly contained in the regular representation. The criterion concerns the regular representation itself; weak containment in an arbitrary representation is not a definition of amenability.

References

Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan's Property (T), Cambridge University Press, 2008. Appendix G DOI record. Relevant: §G.3 on weak containment and amenability.
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, 1988. AMS DOI record. Relevant: Reiter conditions and amenability of locally compact groups.