TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)audited 2026-10-08

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Hulanicki–Reiter weak containment criterion for amenability

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ . Let $1_{G}$ be the trivial unitary representation on $C$ with its standard inner product, given by $1_{G} (g) z = z$ , and let $λ_{G}$ be the left regular representation on $L^{2} (G)$ (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful). Then $G$ is amenable (Amenable locally compact group) if and only if $1_{G} ≺ λ_{G}$ (Weak containment of unitary representations). Equivalently, $G$ is amenable if and only if $λ_{G}$ almost has invariant unit vectors: for every compact $Q \subseteq G$ and every $ε > 0$ there is a unit vector $ξ \in L^{2} (G)$ with $\sup_{x \in Q} ∥ λ_{G} (x) ξ - ξ ∥_{2} < ε$ .

Facts & Assumptions

Given: AC, a locally compact Hausdorff group $G$ , a fixed left Haar measure $μ$ , and the representations $1_{G}$ on $C$ and $λ_{G}$ on $L^{2} (G)$ .

[A1]

AC is assumed in the choice-function form (The Axiom of Choice).

[F1]

Under AC, amenability of a locally compact Hausdorff group is equivalent to Reiter's condition (P1) (Amenability is equivalent to Reiter's condition (P1)).

[F2]

Reiter (P1) means that for every compact $Q$ and every $δ > 0$ there is $f \in L^{1} (G)$ with $f \geq 0$ , $∥ f ∥_{1} = 1$ , and $\sup_{x \in Q} ∥ L_{x} f - f ∥_{1} \leq δ$ (Reiter's condition (P1)).

[F3]

The left regular action is $λ_{G} (x) ξ (y) = ξ (x^{- 1} y)$ ; it is a unitary representation, and under AC it is strongly continuous (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful).

[F4]

$L^{1} (G)$ and $L^{2} (G)$ are complex almost-everywhere classes with $∥ f ∥_{1} = \int_{G} ∣ f ∣ d μ$ and $∥ ξ ∥_{2}^{2} = \int_{G} ∣ ξ ∣^{2} d μ$ ; the $L^{2}$ pairing is the integral pairing (Complex Haar L^p spaces and compactly supported functions, The complex $L^{2}$ pairing on equivalence classes).

[F5]

For vectors $u, v$ in a complex inner-product space, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F6]

Complex modulus is subadditive and satisfies $∣ ∣ a ∣ - ∣ b ∣ ∣ \leq ∣ a - b ∣$ for $a, b \in C$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F7]

Weak containment $π ≺ ρ$ means that every diagonal coefficient of $π$ is uniformly approximable on compact subsets by finite sums of diagonal coefficients of $ρ$ (Weak containment of unitary representations).

[F8]

Under AC, $1_{G} ≺ π$ is equivalent to existence of unit vectors in $π$ that are arbitrarily invariant on each compact subset (Weak containment of the trivial representation and almost invariant vectors).

[F9]

With the standard inner product on $C$ , the diagonal coefficient of $1_{G}$ at $z \in C$ is the constant function $∣ z ∣^{2}$ ; this follows from $1_{G} (g) z = z$ and the matrix-coefficient definition (Real and complex inner-product spaces and their induced length, Matrix coefficient of a unitary representation).

[F10]

Amenability is existence of a left-invariant mean on $L^{\infty} (G)$ (Amenable locally compact group).

[F11]

A strongly continuous unitary representation has continuous orbit maps; the trivial action on $C$ is constant, hence strongly continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F12]

Left Haar measure is left invariant, and the left regular action uses $(L_{x} f) (y) = f (x^{- 1} y)$ (Left Haar integral and left Haar measure, Left and right regular unitary representations of an LCH group).

Proof

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , and its left regular representation $λ_{G}$ .

Proof technique: direct.

1.1F1F10given

Suppose $G$ is amenable in the sense of [F10]. By [F1], $G$ satisfies Reiter (P1).

1.2F2F3F4F12algebra

Fix compact $Q \subseteq G$ and $ε > 0$ . By [F2] choose $f \in L^{1} (G)$ with $f \geq 0$ , $∥ f ∥_{1} = 1$ , and $\sup_{x \in Q} ∥ L_{x} f - f ∥_{1} \leq ε^{2} / 2$ . Set $g : = \sqrt{f} \in L^{2} (G)$ ; then $∥ g ∥_{2} = 1$ . For $a, b \geq 0$ with $a \neq b$ , $∣ \sqrt{a} - \sqrt{b} ∣^{2} = ∣ a - b ∣ ∣ \sqrt{a} - \sqrt{b} ∣ / (\sqrt{a} + \sqrt{b}) \leq ∣ a - b ∣$ , and for $a = b$ both sides are zero. Thus for every $x \in Q$ , $∥ λ_{G} (x) g - g ∥_{2}^{2} = \int_{G} ∣ \sqrt{f (x^{- 1} y)} - \sqrt{f (y)} ∣^{2} d μ (y) \leq ∥ L_{x} f - f ∥_{1} \leq ε^{2} / 2$ , so $\sup_{x \in Q} ∥ λ_{G} (x) g - g ∥_{2} < ε$ .

1.3F3F5F7F9F11given

Assume that $λ_{G}$ almost has invariant unit vectors. Let $Q$ be compact, $τ > 0$ , and let $z \in C$ with $c : = ∣ z ∣^{2}$ . If $c = 0$ , the coefficient of the zero vector is exactly $0$ . If $c > 0$ , choose a unit vector $ξ \in L^{2} (G)$ with $\sup_{x \in Q} ∥ λ_{G} (x) ξ - ξ ∥_{2} < τ / c$ and put $η : = \sqrt{c} ξ$ . By [F5], $∣ 1 - ⟨ λ_{G} (x) ξ, ξ ⟩ ∣ = ∣ ⟨ ξ - λ_{G} (x) ξ, ξ ⟩ ∣ \leq ∥ λ_{G} (x) ξ - ξ ∥_{2}$ , so $\sup_{x \in Q} ∣ c - ⟨ λ_{G} (x) η, η ⟩ ∣ < τ$ . By [F9] every diagonal coefficient of $1_{G}$ is such a constant $c \geq 0$ , and each approximant used here is one coefficient of $λ_{G}$ ; thus [F7] gives $1_{G} ≺ λ_{G}$ .

1.4A1F3F8given

If $1_{G} ≺ λ_{G}$ , [F8] gives almost invariant unit vectors for $λ_{G}$ on every compact subset.

2.1F2F3F4F5F6F12step 1.4algebra

Fix compact $Q \subseteq G$ and $ε > 0$ . By step 1.4 choose a unit vector $ξ \in L^{2} (G)$ with $\sup_{x \in Q} ∥ λ_{G} (x) ξ - ξ ∥_{2} < ε / 2$ . Set $f : = ∣ ξ ∣^{2} \in L^{1} (G)$ , so $f \geq 0$ and $∥ f ∥_{1} = ∥ ξ ∥_{2}^{2} = 1$ . Since $L_{x} f = ∣ λ_{G} (x) ξ ∣^{2}$ , [F6] gives $∣ ∣ λ_{G} (x) ξ ∣^{2} - ∣ ξ ∣^{2} ∣ \leq ∣ λ_{G} (x) ξ - ξ ∣ (∣ λ_{G} (x) ξ ∣ + ∣ ξ ∣)$ pointwise. By [F5], $∥ L_{x} f - f ∥_{1} \leq ∥ λ_{G} (x) ξ - ξ ∥_{2} ∥ ∣ λ_{G} (x) ξ ∣ + ∣ ξ ∣ ∥_{2}$ ; the pointwise estimate $(∣ a ∣ + ∣ b ∣)^{2} \leq 2 (∣ a ∣^{2} + ∣ b ∣^{2})$ and unitarity [F3] give $∥ ∣ λ_{G} (x) ξ ∣ + ∣ ξ ∣ ∥_{2} \leq 2$ . Therefore $∥ L_{x} f - f ∥_{1} \leq 2 ∥ λ_{G} (x) ξ - ξ ∥_{2} < ε$ for all $x \in Q$ , so Reiter (P1) holds.

3.1A1F1F3F8F10step 1.1step 1.2step 1.3step 1.4step 2.1∎

Reiter (P1) implies amenability in the sense of [F10] by [F1], while steps 1.1–2.1 give amenability implies $1_{G} ≺ λ_{G}$ and $1_{G} ≺ λ_{G}$ implies Reiter (P1). By [F8], weak containment is equivalent to almost invariant unit vectors, proving both formulations in the Statement. AC is used only through the cited suppliers [F1], [F3], and [F8].

Sources

BHV, Kazhdan's Property (T), Appendix G.3, Theorem G.3.2 (Hulanicki–Reiter), printed pp. 456–457, proves amenability iff $1_{G} ≺ λ_{G}$ by converting between Reiter densities and almost-invariant $L^{2}$ vectors. Appendix F.1, Corollary F.1.5 and its proof, printed pp. 423–424, gives the weak-containment/almost-invariant-vector equivalence. Thomas, Lecture 20, slides 17–18 (PDF pp. 17–18), gives the same Reiter-to- $L^{2}$ conversion.

Depends on

Used by

Dependency tree · two levels

75 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (standard reference, not scraped)
Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (standard reference, not scraped)
Anne Thomas, The Banach-Tarski Paradox and Amenability, Lecture 20: Invariant Mean implies Reiter's Property (standard reference, not scraped)