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A UCB-invariant mean yields a topological invariant mean

Statement

Assume AC. Let $G$ be a locally compact Hausdorff group with fixed left Haar measure $μ$ , and let $m$ be a left-invariant mean on the actual-function space $U C B (G)$ of Left-uniformly continuous bounded functions (UCB). Thus $m$ is a positive complex-linear functional with $m (1_{G}) = 1$ and $m (L_{g} ψ) = m (ψ)$ for every $g \in G$ and $ψ \in U C B (G)$ . Fix $f_{0} \in P$ for $P$ of Reiter's condition (P1), and define $\tilde{m} (φ) : = m (f_{0} * φ) (φ \in L^{\infty} (G)),$ where $*$ is the pointwise $L^{1}$ -to- $L^{\infty}$ smoothing of L1 convolution smooths bounded functions into UCB. Then $\tilde{m}$ is a mean on $L^{\infty} (G)$ and is topologically invariant: $\tilde{m} (f * φ) = \tilde{m} (φ) (f \in P, φ \in L^{\infty} (G)) .$

Facts & Assumptions

Given: AC, an LCH group $G$ with fixed left Haar measure $μ$ , a left-invariant mean $m$ on $U C B (G)$ , and the probability densities $P$ .

[A1]

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

[F1]

$U C B (G)$ consists of actual bounded continuous functions; its left-translation orbit is sup-norm continuous, translations preserve UCB, and its sup norm agrees with the embedded $L^{\infty}$ norm (Left-uniformly continuous bounded functions (UCB)).

[F2]

The smoothing formula $f * φ (x) = \int f (y) φ (y^{- 1} x) d μ (y)$ is an actual bounded UCB function independent of representatives, with sup bound, left-equivariance and associativity $(f * b) * φ = f * (b * φ)$ for $f, b \in L^{1}$ , $φ \in L^{\infty}$ (L1 convolution smooths bounded functions into UCB).

[F3]

$C_{c} (G)$ is dense in $L^{1} (G)$ under AC. The extended convolution is a continuous bilinear operation agreeing with the Cc formula and satisfying $∥ u * v ∥_{1} \leq ∥ u ∥_{1} ∥ v ∥_{1}$ ; the Cc convolution kernel is continuous and compactly supported (Completeness of the complex Haar L1 and L2 spaces and density of Cc, Convolution on L1 of a locally compact group, Submultiplicativity of convolution in the L1 norm, Compactly supported convolution on a group, Convolution preserves compact support and is associative).

[F4]

For Cc kernels, Fubini interchanges the compactly supported Radon integrals under AC; left Haar invariance gives $\int v (y^{- 1} x) d μ (x) = \int v d μ$ (Compactly supported kernels admit commuting radon integrals, Left Haar integral and left Haar measure).

[F5]

The positive probability approximate-identity net $(e_{U})$ lies in $C_{c} (G) \cap P$ and satisfies $∥ f * e_{U} - f ∥_{1} \to 0$ for every $f \in L^{1} (G)$ under AC (L1 group algebras have a contractively bounded approximate identity, Reiter's condition (P1)).

[F6]

The integral is linear, satisfies $∣ \int u ∣ \leq \int ∣ u ∣$ , and is monotone and positively homogeneous on nonnegative measurable functions; a nonnegative function with zero integral vanishes almost everywhere (Integrable real and complex functions, and their integrals, The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F7]

On a compact subset of a Hausdorff space, a finite open cover can be disjointified into a finite Borel partition; samples can be chosen from its finitely many nonempty cells (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, The Borel sigma-algebra of a topological space, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, Every natural-number-indexed list of nonempty sets has a choice function on its family of values).

Proof

technique · direct

1.1A1F3F7algebra

First, $m$ has norm one. For a real-valued $ψ \in U C B (G)$ , $∥ ψ ∥_{\sup} 1_{G} \pm ψ \geq 0$ , so positivity and $m (1_{G}) = 1$ imply $m (ψ) \in R$ and $∣ m (ψ) ∣ \leq ∥ ψ ∥_{\sup}$ . For complex $ψ$ , choose $α$ with $∣ α ∣ = 1$ and $α m (ψ) = ∣ m (ψ) ∣$ ; conjugation preserves UCB, so $Re (α ψ) \in U C B (G)$ and $∣ m (ψ) ∣ = m (Re (α ψ)) \leq ∥ ψ ∥_{\sup}$ by positivity. Testing at $1_{G}$ gives $∥ m ∥ = 1$ . Now fix $f, b \in P$ . By [F3] choose Cc approximants $u_{n}, v_{n}$ to $f, b$ in $L^{1}$ , and replace them by $∣ u_{n} ∣, ∣ v_{n} ∣$ ; since $f, b \geq 0$ and $∣ ∣ u_{n} ∣ - f ∣ \leq ∣ u_{n} - f ∣$ , these remain convergent nonnegative Cc approximants.

2.1A1F1F2F3F7constructstep 1.1

Fix $f \in P$ and $ψ \in U C B (G)$ . Given $ε > 0$ , choose $u \in C_{c} (G)$ with $∥ f - u ∥_{1} < ε$ and put $K = supp u$ ; then $\int_{G ∖ K} f d μ < ε$ . Since $y \mapsto L_{y} ψ$ is sup-norm continuous and $K$ is compact, a finite open cover of $K$ gives a finite Borel partition $E_{1}, \dots, E_{N}$ of $K$ and sample points $y_{j} \in K$ with $∥ L_{y} ψ - L_{y_{j}} ψ ∥_{\sup} < ε$ for $y \in E_{j}$ . Set $S = \sum_{j} (\int_{E_{j}} f d μ) L_{y_{j}} ψ$ . The pointwise smoothing formula in [F2] gives $∥ f * ψ - S ∥_{\sup} \leq ε + ε ∥ ψ ∥_{\sup}$ , from the partition error on $K$ and the tail outside $K$ . Invariance and linearity of $m$ give $m (S) = m (ψ) \int_{K} f d μ$ , hence $∣ m (S) - m (ψ) ∣ \leq ε ∥ ψ ∥_{\sup}$ . By step 1.1, $∣ m (f * ψ) - m (ψ) ∣ \leq ε (1 + 2 ∥ ψ ∥_{\sup})$ ; letting $ε ↓ 0$ proves $m (f * ψ) = m (ψ)$ . This uses finite Borel partitions and pointwise scalar integrals, with no Bochner measurability or separability assumption.

2.2F2F6step 1.1

Define $\tilde{m} (φ) = m (f_{0} * φ)$ . By [F2], $\tilde{m}$ is complex-linear and $∣ \tilde{m} (φ) ∣ \leq ∥ f_{0} * φ ∥_{\sup} \leq ∥ φ ∥_{\infty}$ by step 1.1. If $φ \geq 0$ , the pointwise smoothing formula gives $f_{0} * φ \geq 0$ , hence $\tilde{m} (φ) \geq 0$ . Also $f_{0} * 1_{G} = 1_{G}$ because $\int f_{0} = 1$ , so $\tilde{m} (1_{G}) = 1$ . Therefore $\tilde{m}$ is a mean on $L^{\infty} (G)$ .

2.3F3F4F6step 1.1

For nonnegative $u, v \in C_{c} (G)$ , the compactly supported convolution formula gives $u * v \geq 0$ . Its integral is $\int_{x} \int_{y} u (y) v (y^{- 1} x) d μ (y) d μ (x) = \int_{y} u (y) \int_{z} v (z) d μ (z) d μ (y) = (\int u) (\int v)$ by [F4] and the substitution $x = y z$ . Applying this to the approximants fixed in step 1.1 gives convolution masses tending to $(\int f) (\int b) = 1$ .

3.1F2F5step 1.1step 2.1

Let $(e_{U})$ be the net of [F5]. For any $f \in P$ and $φ \in L^{\infty} (G)$ , $e_{U} * φ \in U C B (G)$ by [F2], so step 2.1 and associativity in [F2] give $m ((f * e_{U}) * φ) = m (f * (e_{U} * φ)) = m (e_{U} * φ)$ . Also [F2] gives $∥ (f * e_{U}) * φ - f * φ ∥_{\sup} \leq ∥ f * e_{U} - f ∥_{1} ∥ φ ∥_{\infty} \to 0$ by [F5]. Since $m$ is bounded by step 1.1, $m (f * φ) = \lim_{U} m (e_{U} * φ)$ , independent of $f \in P$ . Hence $m (f * φ) = m (f_{0} * φ)$ for all $f, f_{0} \in P$ .

3.2F3F6step 1.1step 2.3

The extension bound in [F3] gives $u_{n} * v_{n} \to f * b$ in $L^{1}$ for the nonnegative Cc approximants from step 1.1: the difference is bounded by $∥ u_{n} - f ∥_{1} ∥ v_{n} ∥_{1} + ∥ f ∥_{1} ∥ v_{n} - b ∥_{1}$ . Since each $u_{n} * v_{n}$ is nonnegative, $∥ Im (f * b) ∥_{1}$ and $∥ (Re (f * b))^{-} ∥_{1}$ are bounded by $∥ u_{n} * v_{n} - f * b ∥_{1}$ and tend to zero; [F6] implies $f * b \geq 0$ almost everywhere. Continuity of integration on $L^{1}$ , also in [F6], gives $\int f * b = \lim_{n} (\int u_{n}) (\int v_{n}) = 1$ by step 2.3. Thus $f * b \in P$ .

4.1F2step 3.1step 3.2step 2.2∎

For $f \in P$ and $φ \in L^{\infty} (G)$ , associativity in [F2] gives $\tilde{m} (f * φ) = m (f_{0} * (f * φ)) = m ((f_{0} * f) * φ)$ . Step 3.2 gives $f_{0} * f \in P$ , so the kernel independence proved in step 3.1 makes this value $m (f_{0} * φ) = \tilde{m} (φ)$ . Every argument of $m$ here is a smoothed UCB function by [F2]; no value of $m$ on an arbitrary unsmoothed $L^{\infty}$ input is used. Together with step 2.2 this proves that $\tilde{m}$ is a topological invariant mean.

Remark

Step 3.2 also proves that the extended $L^{1}$ convolution preserves probability densities: for all $f, b \in P$ , one has $f * b \in P$ . The local proof uses nonnegative compactly supported approximants, compact-support Radon integration, and convergence in the $L^{1}$ convolution norm.

Sources

BHV, Kazhdan's Property (T), Appendix G, §G.3, proof of Theorem G.3.1, (i) to (ii), printed pp. 453–454, proves the identity $m (f * φ) = m (φ)$ for UCB inputs, compares probability kernels using a positive approximate identity, and defines $\tilde{m} (φ) = m (f_{0} * φ)$ . Thomas, Lecture 20, slides 11–14, gives the same construction. The local proof justifies the compact-partition approximation and uses the right-L1 estimate $∥ f * e_{U} - f ∥_{1}$ with the smoothing bound; it makes no sup-norm approximate- identity claim for arbitrary L-infinity inputs.

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Sources

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)