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Raikov: compact-open and weak star topologies agree on normalized positive type functions

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group with a fixed left Haar measure and let $P_{1} (G)$ be the set of continuous functions of positive type $φ$ with $φ (e) = 1$ (Continuous positive-type functions and normalization), viewed in the unit ball of $L^{\infty} (G; C)$ . Here this means complex essentially bounded measurable functions modulo equality almost everywhere, paired with the complex Haar $L^{1} (G)$ by $[u] \mapsto (f \mapsto \int f u)$ . Each class defines a bounded functional on $(L^{1} (G; C))$ by this pairing. Its restriction to $P_{1} (G)$ is injective; no identification of the entire $L^{\infty}$ space with the dual is required. On $P_{1} (G)$ the weak-* topology $σ (L^{\infty}, L^{1})$ , that is, the topology of convergence of $\int_{G} f φ_{i}$ for every $f \in L^{1} (G)$ , coincides with the topology of uniform convergence on compact subsets of $G$ : a net $(φ_{i}) \subseteq P_{1} (G)$ satisfies $\int_{G} f φ_{i} \to \int_{G} f φ$ for every $f \in L^{1} (G)$ if and only if $φ_{i} \to φ$ uniformly on every compact $Q \subseteq G$ .

Facts & Assumptions

Given: AC; an LCH group $G$ with fixed left Haar measure $μ$ ; the set $P_{1} (G)$ ; a net $(φ_{i})_{i \in I} \subseteq P_{1} (G)$ and $φ \in P_{1} (G)$ .

[A1]

A continuous positive-type function is defined by the positive semidefiniteness of the matrices $(φ (g_{j}^{- 1} g_{k}))$ ; for $φ \in P_{1} (G)$ the $2 \times 2$ matrix with entries $φ (e) = 1, φ (g), φ (g^{- 1}), 1$ is positive semidefinite, so $φ (g^{- 1}) = \overline{φ (g)}$ and $∣ φ (g) ∣ \leq 1$ for every $g$ (Continuous positive-type functions and normalization).

[A2]

Translation estimates: if $ψ$ has a GNS triple $(π_{ψ}, H_{ψ}, ξ)$ with $∥ ξ ∥ = 1$ and $ψ (g) = ⟨ π_{ψ} (g) ξ, ξ ⟩$ , then $∣ ψ (x) - ψ (y) ∣^{2} \leq 2 (1 - Re ψ (y^{- 1} x))$ and $2 (1 - Re ψ (g)) = ∥ π_{ψ} (g) ξ - ξ ∥^{2}$ for all $x, y, g$ (Translation estimates for continuous positive type functions). Every $ψ \in P_{1} (G)$ is the diagonal coefficient of a strongly continuous unitary representation with a cyclic unit vector with $ψ (e) = 1$ , namely its GNS triple (GNS construction for a continuous positive-type function, Matrix coefficient of a unitary representation, Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[A3]

$L^{1} (G)$ is a Banach space, $C_{c} (G) \subseteq L^{1} (G)$ is dense, and left translations $L_{x} f (y) = f (x^{- 1} y)$ are isometric with $x \mapsto L_{x} f$ continuous (Complex Haar L^p spaces and compactly supported functions, Completeness of the complex Haar L1 and L2 spaces and density of Cc, Left Haar integral and left Haar measure, Strong continuity of left and modular right translations on L1 and L2).

[A4]

For bounded measurable $ψ$ with $∣ ψ ∣ \leq 1$ , the map $f \mapsto \int_{G} f ψ d μ$ is a bounded linear functional on $L^{1} (G)$ of norm at most $1$ . A continuous function $u$ not identically zero has a nonzero pairing: choose a compact neighbourhood $V$ inside an open set where $∣ u ∣ > c > 0$ , and use $f = 1_{V} \overline{u} \in L^{1} (G)$ , giving $\int f u = \int_{V} ∣ u ∣^{2} > 0$ . Such a $V$ has finite positive Haar measure (Haar measure is positive on nonempty open sets and finite on compact sets). Thus the pairing embeds $P_{1} (G)$ faithfully in the dual, and the restricted weak-* topology is the topology of the stated evaluations (Weak star convergence, Directed preorders and nets).

[A5]

Cauchy–Schwarz for a probability measure: $∣ \int u d ν ∣^{2} \leq \int ∣ u ∣^{2} d ν$ when $ν \geq 0$ has total mass one (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A6]

Every open neighbourhood $W$ of a point in an LCH space contains a compact neighbourhood of that point. To see this, choose a compact neighbourhood $K$ with open $O \subseteq K$ containing the point. Complete regularity supplies a continuous $f : X \to [0, 1]$ equal to $1$ there and zero outside $W \cap O$ . Then $V = K \cap {f \geq 1 / 2}$ is compact, is contained in $W$ , and contains the open set ${f > 1 / 2}$ . Complete regularity under DC is available under AC (Under dependent choice a locally compact Hausdorff space is completely regular, hence Tychonoff, AC implies DC implies countable choice).

Proof

technique · direct

Given: AC, an LCH group $G$ with left Haar measure, a net $(φ_{i}) \subseteq P_{1} (G)$ and $φ \in P_{1} (G)$ .

1.1A1

For every $ψ \in P_{1} (G)$ one has $∣ ψ (g) ∣ \leq 1$ and $ψ (g^{- 1}) = \overline{ψ (g)}$ : the matrix $(\begin{array}{cc} 1 & ψ (g) \\ ψ (g^{- 1}) & 1 \end{array})$ is positive semidefinite by [A1], so it is Hermitian with nonnegative determinant.

1.2A1A2A3A5

Averaging estimate. Let $V \subseteq G$ be a compact identity neighbourhood with $μ (V) > 0$ , put $f : = μ (V)^{- 1} 1_{V} \in L^{1} (G)$ , and for $ψ \in P_{1} (G)$ define $A ψ (x) : = \int_{G} f (h) ψ (x h) d h$ . Then $\sup_{x \in G} ∣ A ψ (x) - ψ (x) ∣ \leq (2 (1 - Re \int_{G} f ψ))^{1 / 2}$ . Indeed, the change of variables $g = x h$ and left invariance of Haar measure give $A ψ (x) = \int_{G} f (x^{- 1} g) ψ (g) d g$ , and [A2] together with the $2 \times 2$ case of [A1] gives $∣ ψ (x h) - ψ (x) ∣ \leq (2 (1 - Re ψ (h)))^{1 / 2}$ ; hence $∣ A ψ (x) - ψ (x) ∣ \leq \int f (h) (2 (1 - Re ψ (h)))^{1 / 2} d h$ , and Cauchy–Schwarz for the probability measure $f d h$ of total mass one, [A5], bounds this by $(2 \int f (h) (1 - Re ψ (h)) d h)^{1 / 2} = (2 (1 - Re \int f ψ))^{1 / 2}$ by linearity of the integral.

2.1A3A4step 1.1

Uniformity on compact sets. Suppose $\int_{G} u φ_{i} \to \int_{G} u φ$ for every $u \in L^{1} (G)$ . Then for every compact $Q \subseteq G$ and every $f \in L^{1} (G)$ , $\sup_{x \in Q} ∣ \int_{G} (L_{x} f) (φ_{i} - φ) ∣ \to 0$ . Indeed, $K : = {L_{x} f : x \in Q}$ is a compact subset of $L^{1} (G)$ by [A3]; let $M : = \sup_{i} ∥ φ_{i} ∥_{\infty} \lor ∥ φ ∥_{\infty} \leq 1$ by step 1.1. Given $ϵ > 0$ , cover $K$ by finitely many balls $B (u_{j}, ϵ / (4 M + 1))$ , $j = 1, \dots, m$ ; for each $j$ the assumed convergence gives $∣ \int u_{j} (φ_{i} - φ) ∣ < ϵ / 2$ eventually, and a common bound $i_{0}$ works for all $j$ ; for $i \geq i_{0}$ and any $x$ with $L_{x} f \in B (u_{j}, ϵ / (4 M + 1))$ one has $∣ \int (L_{x} f - u_{j}) (φ_{i} - φ) ∣ \leq ∥ L_{x} f - u_{j} ∥_{1} ∥ φ_{i} - φ ∥_{\infty} < ϵ / 2$ , so the sum is $< ϵ$ uniformly over $x \in Q$ .

2.2A3step 1.1

Compact-open convergence implies weak-* convergence. If $φ_{i} \to φ$ uniformly on compacta, then $\int_{G} f (φ_{i} - φ) \to 0$ for every $f \in L^{1} (G)$ : given $ϵ > 0$ , [A3] and absolute continuity of the integral provide a compact $Q$ with $\int_{G ∖ Q} ∣ f ∣ < ϵ / 4$ ; by step 1.1 both $φ_{i}$ and $φ$ are bounded by $1$ , so $∣ \int f (φ_{i} - φ) ∣ \leq ∥ f ∥_{1} \sup_{Q} ∣ φ_{i} - φ ∣ + 2 \int_{G ∖ Q} ∣ f ∣ < ϵ$ once $\sup_{Q} ∣ φ_{i} - φ ∣ < ϵ / (2 ∥ f ∥_{1} + 2)$ .

3.1A1A3A6step 1.2step 2.1

Weak-* convergence implies compact-open convergence. Assume $\int_{G} f φ_{i} \to \int_{G} f φ$ for every $f \in L^{1} (G)$ , let $Q \subseteq G$ be compact and let $ϵ > 0$ . By continuity of $φ$ at $e$ and $φ (e) = 1$ , choose, using [A6], a compact identity neighbourhood $V$ with $\sup_{V} ∣ 1 - φ ∣ < δ$ for a small $δ > 0$ to be fixed below, and let $f = μ (V)^{- 1} 1_{V}$ . For all $i$ eventually, $1 - Re \int f φ_{i} < 2 δ$ , because $\int f φ_{i} \to \int f φ$ and $1 - Re \int f φ = \int f (1 - Re φ) \leq \sup_{V} ∣ 1 - φ ∣ < δ$ . Hence by step 1.2, $\sup_{x \in G} ∣ A φ_{i} (x) - φ_{i} (x) ∣ \leq 2 \sqrt{δ}$ and $\sup_{x \in G} ∣ A φ (x) - φ (x) ∣ \leq \sqrt{2 δ}$ eventually (for $φ$ itself directly, for $φ_{i}$ once the displayed inequality holds). By step 2.1, $\sup_{x \in Q} ∣ A φ_{i} (x) - A φ (x) ∣ \leq ϵ / 3$ eventually, because $A ψ (x) = \int (L_{x} f) ψ$ as computed in step 1.2. Therefore eventually $\sup_{Q} ∣ φ_{i} - φ ∣ \leq 2 \sqrt{δ} + ϵ / 3 + \sqrt{2 δ}$ ; taking $δ$ so small that $2 \sqrt{δ} + \sqrt{2 δ} < ϵ / 3$ gives $\sup_{Q} ∣ φ_{i} - φ ∣ < ϵ$ .

4.1A1A3step 2.2step 3.1∎

Steps 3.1 and 2.2 prove the two implications for an arbitrary net, hence the two topologies on $P_{1} (G)$ coincide; no compactness theorem for $P_{1} (G)$ and no unimodularity is used, and the averaging in step 1.2 is matched with left translation in the $L^{1}$ pairing. The Axiom of Choice is inherited from the Haar and translation suppliers of [A1]–[A5]; the estimates themselves are choice-free (The Axiom of Choice).

Depends on

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72 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) (Cambridge University Press 2008; author-hosted complete text) (standard reference, not scraped)
Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019) (standard reference, not scraped)