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Continuous characters separate points of an LCA group

Statement

Assume the Axiom of Choice (The Axiom of Choice) and Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $G$ be a locally compact Hausdorff abelian group (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Topological group: multiplication and inversion are continuous) and let $x \in G$ with $x \neq 0$ . Then there is $γ \in \hat{G}$ with $γ (x) \neq 1$ . Equivalently, the evaluation map $Φ (x) (γ) : = γ (x)$ is injective on $G$ .

Facts & Assumptions

Given: A locally compact Hausdorff abelian group $G$ and a point $x \in G$ with $x \neq 0$ .

[F1]

In a Hausdorff space distinct points have disjoint open neighbourhoods; in a locally compact Hausdorff space every point has a neighbourhood basis of open sets with compact closure. In a topological group addition is continuous and translations and inversion are homeomorphisms. (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, Topological group: multiplication and inversion are continuous, Left and right translations and inversion in a topological group are homeomorphisms)

[F2]

If $K \subseteq U$ with $K$ compact and $U$ open in a locally compact Hausdorff space $X$ , then under Dependent Choice there is $f \in C_{c} (X)$ with $1_{K} \leq f \leq 1_{U}$ . (LCH Urysohn cutoff)

[F3]

For $g \in C_{c} (G)$ put $\tilde{g} (x) : = \overline{g (- x)}$ . Then $g * \tilde{g} \in C_{c} (G)$ is continuous with compact support, is positive definite, and $(g * \tilde{g}) (0) = \int_{G} ∣ g ∣^{2} d m_{G} \geq 0$ ; the convolution is $(f * g) (x) = \int_{G} f (y) g (x - y) d m_{G} (y)$ and $C_{c} (G) \subseteq L^{1} (G) \cap L^{2} (G)$ . A left Haar measure is strictly positive on nonzero nonnegative compactly supported functions, so $\int_{G} ∣ g ∣^{2} > 0$ whenever $g \equiv̸ 0$ . (Positive convolution squares form a dense inversion core, L^1 of an LCA group is a commutative Banach star algebra under convolution, Compact support, $C_{c} (X)$ , and $C_{0} (X)$ , Haar measure is positive on nonempty open sets and finite on compact sets, Left Haar integral and left Haar measure)

[F4]

Bochner's theorem: a continuous function $ϕ : G \to C$ is positive definite if and only if there is a unique finite positive Radon measure $μ$ on $\hat{G}$ with $ϕ (x) = \int_{\hat{G}} γ (x) d μ (γ)$ for all $x$ , and then $μ (\hat{G}) = ϕ (0)$ . (Bochner's theorem for LCA groups, Positive definite functions on an abelian group, Fourier-Stieltjes transforms of positive measures are continuous positive definite)

[F5]

For $g \in C_{c} (G)$ and $y \in G$ one has $\overline{g (y - x)} = \tilde{g} (x - y)$ , and $x \notin S - S$ is equivalent to $S \cap (x + S) = \emptyset$ . (Topological group: multiplication and inversion are continuous, Left and right translations and inversion in a topological group are homeomorphisms)

Proof

1.1F1

Because $x \neq 0$ and $G$ is Hausdorff, addition is continuous at $(0, 0)$ and $G ∖ {x}$ is open, so there are open neighbourhoods $W_{1}, W_{2}$ of $0$ with $W_{1} + W_{2} \subseteq G ∖ {x}$ ; replacing them by their intersections with their negatives and with each other, we obtain a symmetric open neighbourhood $W$ of $0$ with $(W + W) \cap {x} = \emptyset$ .

2.1F1F5step 1.1

Choose a symmetric compact neighbourhood $S$ of $0$ with $S \subseteq W$ : a neighbourhood basis of open sets with compact closure at $0$ supplies an open $V$ with $0 \in V \subseteq {c l}_{G} (V) \subseteq W$ , and $S : = {c l}_{G} (V) \cap (- {c l}_{G} (V))$ is compact and symmetric with $0 \in S \subseteq W$ . Then $S - S \subseteq W + W$ , so $x \notin S - S$ ; hence $S \cap (x + S) = \emptyset$ , since $s = x + s^{'}$ would give $x = s - s^{'} \in S - S$ .

3.1F2F3step 2.1

Apply the cutoff of [F2] with $K = {0}$ and $U = {i n t}_{G} (S)$ to obtain $g \in C_{c} (G)$ with $g \geq 0$ , $g (0) = 1$ and $supp g \subseteq S$ . Then $h : = g * \tilde{g}$ is continuous with compact support, positive definite, and $h (0) = \int_{G} ∣ g ∣^{2} d m_{G} > 0$ because $g \equiv̸ 0$ and $g \geq 0$ .

4.1F3F5step 2.1step 3.1

For this $h$ one has $h (x) = 0$ : by the convolution formula and $\tilde{g} (x - y) = \overline{g (y - x)}$ from [F5], $h (x) = \int_{G} g (y) \overline{g (y - x)} d m_{G} (y)$ , and the integrand vanishes identically because $g (y) \neq 0$ forces $y \in S$ while $g (y - x) \neq 0$ forces $y \in x + S$ , and $S \cap (x + S) = \emptyset$ .

5.1F4step 3.1step 4.1

By Bochner's theorem [F4] there is a unique finite positive Radon measure $μ$ on $\hat{G}$ with $h (x) = \int_{\hat{G}} γ (x) d μ (γ)$ for all $x \in G$ and $μ (\hat{G}) = h (0) > 0$ . If $γ (x) = 1$ for every $γ \in \hat{G}$ , then $h (x) = \int_{\hat{G}} 1 d μ (γ) = μ (\hat{G}) = h (0) > 0$ , contradicting $h (x) = 0$ from step 4.1. Hence some $γ \in \hat{G}$ satisfies $γ (x) \neq 1$ .

6.1step 5.1∎

Since $x \neq 0$ was arbitrary, continuous characters separate points of $G$ . Equivalently the evaluation map is injective: if $Φ (x) = Φ (x^{'})$ then $γ (x - x^{'}) = γ (x) γ (- x^{'}) = γ (x) \overline{γ (x^{'})} = 1$ for every $γ \in \hat{G}$ , the separation result forces $x - x^{'} = 0$ , that is $x = x^{'}$ . Conversely, if $Φ$ is injective and $x \neq 0$ , then $Φ (x) \neq Φ (0)$ , so some character has $γ (x) \neq 1$ .

Depends on

Used by

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Sources

Lynn H. Loomis, Introduction to Abstract Harmonic Analysis, D. Van Nostrand, 1953 (Harvard-hosted full scan) (standard reference, not scraped)
T. W. Koerner, Topological Groups (author lecture notes) (standard reference, not scraped)