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An invariant mean produces a Reiter net

Statement

Assume AC. Suppose $G$ admits a left-invariant mean on $U C B (G)$ ; this holds in particular when $G$ is amenable in the sense of Amenable locally compact group. Then $G$ satisfies Reiter's condition (P1) (Reiter's condition (P1)). Consequently every amenable locally compact group satisfies (P1).

Facts & Assumptions

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , and a left-invariant mean on actual bounded uniformly continuous functions $U C B (G)$ .

[A1]

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

[F1]

$P = {f \in L^{1} (G) : f \geq 0, ∥ f ∥_{1} = 1}$ is convex and $L_{x} P = P$ for every $x \in G$ ; (P1) requires a density with the compact-test defect $Δ_{Q} (f) \leq ε$ (Reiter's condition (P1)).

[F2]

$U C B (G)$ consists of actual bounded continuous functions, is translation invariant, and its class map into complex $L^{\infty} (G)$ is an isometric embedding (Left-uniformly continuous bounded functions (UCB)).

[F3]

Amenability supplies a positive complex-linear unital left-invariant mean on complex $L^{\infty} (G)$ (Amenable locally compact group).

[F4]

Under AC, a left-invariant mean on $U C B (G)$ yields a topological invariant mean on $L^{\infty} (G)$ (A UCB-invariant mean yields a topological invariant mean).

[F5]

A topological invariant mean on $L^{\infty} (G)$ yields a net $(g_{j}) \subseteq P$ whose defects $∥ h * g_{j} - g_{j} ∥_{1}$ tend to zero uniformly for $h$ in every norm-compact subset of $P$ (A topological invariant mean yields norm-approximately invariant densities).

[F6]

For each $f \in L^{1} (G)$ , the orbit map $x \mapsto L_{x} f$ is norm-continuous (Strong continuity of left and modular right translations on L1 and L2).

[F7]

Compactness is intrinsic to the subspace, and every ambient open cover of a compact subspace has a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

[F8]

A continuous image of a compact space is compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F9]

Extended $L^{1}$ convolution is bilinear and satisfies $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1}$ ; it agrees with the compact-support convolution on $C_{c} (G)$ (Convolution on L1 of a locally compact group).

[F10]

Extended convolution preserves probability densities: $f * g \in P$ for $f, g \in P$ (A UCB-invariant mean yields a topological invariant mean, Remark).

[F11]

Under AC, $C_{c} (G)$ is dense in $L^{1} (G)$ for a Radon Haar measure (Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F12]

Left Haar measure is left invariant and Radon under the repository convention (Left Haar integral and left Haar measure).

[F13]

For $u, v \in C_{c} (G)$ , $(u * v) (y) = \int_{G} u (z) v (z^{- 1} y) d μ (z)$ (Compactly supported convolution on a group).

[F14]

Left translation preserves $C_{c} (G)$ (Translations preserve compactly supported continuous functions).

Proof

technique · direct

1.1A1F4

By [F4], the given mean on $U C B (G)$ yields a topological invariant mean $\tilde{m}$ on $L^{\infty} (G)$ .

1.2A1F9F11F12F13F14

We first prove left-equivariance of the extended convolution. For $u, v \in C_{c} (G)$ , [F13] and left invariance give, for every $x, y \in G$ , $((L_{x} u) * v) (y) = \int_{G} u (x^{- 1} z) v (z^{- 1} y) d μ (z) = \int_{G} u (w) v (w^{- 1} x^{- 1} y) d μ (w) = (L_{x} (u * v)) (y)$ , where $z = x w$ . Thus $L_{x} (u * v) = (L_{x} u) * v$ in $L^{1} (G)$ . For arbitrary $f, v \in L^{1} (G)$ choose $u_{n}, v_{n} \in C_{c} (G)$ with $u_{n} \to f$ and $v_{n} \to v$ in $L^{1}$ , using [F11]. For fixed $x$ , $L_{x} u_{n} \in C_{c} (G)$ by [F14] and $∥ L_{x} u_{n} - L_{x} f ∥_{1} = ∥ u_{n} - f ∥_{1}$ by [F12]. The convolution bound [F9] then gives $u_{n} * v_{n} \to f * v$ and $(L_{x} u_{n}) * v_{n} \to (L_{x} f) * v$ in $L^{1}$ ; passing the compact-support identity to these limits proves $L_{x} (f * v) = (L_{x} f) * v$ .

2.1A1F5step 1.1

Apply [F5] to $\tilde{m}$ from step 1.1 and fix the resulting net $(g_{j}) \subseteq P$ , with defects converging uniformly on norm-compact subsets of $P$ .

3.1A1F1F6F7F8step 2.1construct

Let $Q \subseteq G$ be compact, $ε > 0$ , and put $Q_{0} : = Q \cup {e}$ . This is compact: for an ambient open cover of $Q_{0}$ , [F7] supplies finitely many members covering $Q$ , and one additional member covers $e$ ; the ambient criterion in [F7] then gives compactness of $Q_{0}$ . The net in step 2.1 shows $P$ is nonempty, so fix $f \in P$ . By [F1], $C : = {L_{x} f : x \in Q_{0}} \subseteq P$ ; by [F6] the orbit map is continuous, and hence $C$ is norm-compact by [F8].

4.1F1F5F10step 1.2step 2.1step 3.1

By [F5] applied to the compact set $C$ , choose an index $j$ such that $∥ h * g_{j} - g_{j} ∥_{1} < ε / 2$ for every $h \in C$ . In particular, $∥ f * g_{j} - g_{j} ∥_{1} < ε / 2$ , since $f = L_{e} f \in C$ . Let $g : = f * g_{j} \in P$ by [F10]. For each $x \in Q_{0}$ , step 1.2 gives $L_{x} g = (L_{x} f) * g_{j}$ , so $∥ L_{x} g - g ∥_{1} \leq ∥ (L_{x} f) * g_{j} - g_{j} ∥_{1} + ∥ f * g_{j} - g_{j} ∥_{1} < ε$ . Therefore $Δ_{Q} (g) \leq ε$ , proving (P1) for arbitrary compact $Q$ and positive $ε$ .

5.1F2F3step 4.1construct∎

If $G$ is amenable, let $ν$ be its mean on $L^{\infty} (G)$ from [F3] and define $m (ψ) : = ν ([ψ])$ for $ψ \in U C B (G)$ . By [F2] this is well-defined, positive, complex-linear and unital; the class map intertwines left translations, so $m$ is left invariant. Applying steps 1.1–4.1 gives (P1).

Sources

BHV, Kazhdan's Property (T), Appendix G.3, Theorem G.3.1 (ii) to (iii), printed p. 455, uses strong continuity to make the translate orbit compact and then applies the uniform approximate-invariance net. Thomas, Lecture 20, slides 17–18 (PDF pp. 17–18), gives the same compact-orbit convolution estimate. Both source proofs take a compact set containing $e$ ; the proof above handles arbitrary compact tests by adjoining $e$ .

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Sources

Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (Cambridge University Press, 2008; author-hosted complete text) (standard reference, not scraped)
Anne Thomas, The Banach-Tarski Paradox and Amenability, Lecture 20: Invariant Mean implies Reiter's Property (University of Sydney Honours lecture notes, 11 October 2012) (standard reference, not scraped)