Reiter condition - Imported Knowlpedia Content

Definition

A locally compact group $G$ satisfies the Reiter condition $P_1$ if there is a net $(u_i)$ in the $L^1$ group algebra such that $u_i\geq0$ , $\lVert u_i\rVert_1=1$ , and, for every compact subset $C\subseteq G$ ,

\sup_{x\in C}\lVert L_xu_i-u_i\rVert_1\longrightarrow0, \qquad (L_xu)(y)=u(x^{-1}y).

The $L^1$ space is formed using a left Haar measure. The condition is independent of its normalization and requires uniform almost invariance on each compact set, not merely pointwise convergence for each fixed group element. Such a net is called a Reiter net.

Equivalence with amenability

Reiter's theorem states that $G$ satisfies $P_1$ exactly when $G$ is amenable Reiter–Stegeman, Chapter 8. Weak-star cluster points of the associated averaging functionals produce an invariant mean. Conversely, an invariant mean yields almost invariant normalized functions through a convexity and approximation argument.

Standard models

If $G$ is compact, the constant density obtained from normalized Haar measure is exactly invariant, so a constant net verifies $P_1$ . For a discrete group, compact sets are finite and the condition becomes the existence of probability masses in $\ell^1(G)$ that are asymptotically invariant under translation on every finite set. Normalized indicators of a Følner net give the basic example Paterson, Chapter 4.

Variants and conventions

Equivalent formulations allow signed functions of norm one after taking absolute values, or require convergence separately for each $x$ together with standard uniformization on compact sets. Conditions $P_p$ for $1\leq p<\infty$ use almost invariant unit vectors in $L^p(G)$ ; they are also equivalent to amenability under the usual locally compact hypotheses Reiter–Stegeman, Chapter 8.

References

Hans Reiter and Jan D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., London Mathematical Society Monographs 22, Oxford University Press, 2000. OUP DOI record. Relevant: Chapter 8 on Reiter conditions and amenability.
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, 1988. AMS DOI record. Relevant: Chapter 4 on analytic characterizations of amenability.