LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

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.

Strong continuity of left and modular right translations on L1 and L2

Statement

Assume AC. Let $G$ be an LCH group with a fixed left Haar measure $μ$ , $Δ_{G}$ its modular function, and for $g \in G$ define $L_{g} f (x) : = f (g^{- 1} x), R_{g} f (x) : = Δ_{G} (g)^{1 / 2} f (x g) .$ Then $g \mapsto L_{g}$ is strongly continuous on $L^{1} (G)$ and on $L^{2} (G)$ , and $g \mapsto R_{g}$ is strongly continuous on $L^{2} (G)$ : for every $f$ in the space and every $ϵ > 0$ there is a neighbourhood $V$ of $e$ in $G$ with $∥ L_{g} f - f ∥_{p} < ϵ$ , respectively $∥ R_{g} f - f ∥_{2} < ϵ$ , for every $g \in V$ — and then, for the general point $g_{0}$ , $∥ L_{g} f - L_{g_{0}} f ∥_{p} \to 0$ and $∥ R_{g} f - R_{g_{0}} f ∥_{2} \to 0$ as $g \to g_{0}$ , neighbourhoods rather than sequences being used throughout since $G$ need not be first countable.

Facts & Assumptions

Given: An LCH group $G$ with a fixed left Haar measure $μ$ , its modular function $Δ_{G}$ , the complex spaces $L^{1} (G), L^{2} (G)$ with norms $∥ \cdot ∥_{p}$ , and AC.

[F1]

For $p \in {1, 2}$ the complex space $L^{p} (G) = L^{p} (G, μ; C)$ consists of the a.e. classes of measurable complex functions with $∥ f ∥_{p} = (\int_{G} ∣ f ∣^{p} d μ)^{1 / p} < \infty$ , and $C_{c} (G) = C_{c} (G; C)$ (Complex Haar L^p spaces and compactly supported functions).

[F2]

Left invariance: $μ (a E) = μ (E)$ for every Borel $E$ and $a \in G$ , so $\int_{G} H (a x) d μ (x) = \int_{G} H (x) d μ (x)$ for nonnegative Borel $H$ (Left Haar integral and left Haar measure); and for each $g$ there is $c (g) > 0$ with $\int_{G} F (x g) d μ (x) = c (g) \int_{G} F d μ (x)$ for every nonnegative Borel $F$ , the scalar being the unique one with that property and equal to $Δ_{G} (g^{- 1})$ (Right translation scales left Haar measure, Uniqueness of left Haar measure up to scale, Modular function of a locally compact group).

[F3]

$Δ_{G}$ is a continuous homomorphism $G \to R_{> 0}$ with $Δ_{G} (e) = 1$ , so $g \mapsto Δ_{G} (g)^{1 / 2}$ is continuous at $e$ with value $1$ , and $Δ_{G} (g) Δ_{G} (g^{- 1}) = 1$ (The modular function is a continuous homomorphism).

[F4]

For $f \in C_{c} (G)$ the maps $a \mapsto L_{a} f$ and $a \mapsto ρ_{a} f$ , where $ρ_{a} f (x) : = f (x a)$ is the unscaled right translate, are continuous in uniform norm at every $a_{0} \in G$ , with all supports contained in one fixed compact set on a neighbourhood of $a_{0}$ ; both translates lie in $C_{c} (G)$ (Translations preserve compactly supported continuous functions). The modular right translate of the Statement is $R_{a} f = Δ_{G} (a)^{1 / 2} ρ_{a} f$ , whose scalar factor is continuous by [F3].

[F5]

$C_{c} (G)$ is dense in $L^{1} (G)$ and in $L^{2} (G)$ (Completeness of the complex Haar L1 and L2 spaces and density of Cc).

[F6]

A left Haar measure is finite on compact sets (Left Haar integral and left Haar measure).

[A1]

AC is assumed in the choice-function form of the cited definition; it is inherited here through the density statement [F5] (The Axiom of Choice).

Proof

technique · direct

1.1

Isometries. Let $f \in L^{1} (G)$ . By [F2] both $\int_{G} ∣ f (g^{- 1} x) ∣ d μ (x) = \int_{G} ∣ f (y) ∣ d μ (y)$ and, since $∣ R_{g} f (x) ∣ = Δ_{G} (g)^{1 / 2} ∣ f (x g) ∣$ , $∥ R_{g} f ∥_{1} = Δ_{G} (g)^{- 1 / 2} ∥ f ∥_{1}$ . For $f \in L^{2} (G)$ the same substitution gives $∥ L_{g} f ∥_{2} = ∥ f ∥_{2}$ , while $∥ R_{g} f ∥_{2}^{2} = \int_{G} Δ_{G} (g) ∣ f (x g) ∣^{2} d μ (x) = Δ_{G} (g) c (g) ∥ f ∥_{2}^{2} = ∥ f ∥_{2}^{2}$ by [F2] and $Δ_{G} (g) Δ_{G} (g^{- 1}) = 1$ from [F3]. So $L_{g}$ is isometric on $L^{1} (G)$ and on $L^{2} (G)$ , and $R_{g}$ is isometric on $L^{2} (G)$ (it scales the $L^{1}$ norm by $Δ_{G} (g)^{- 1 / 2}$ ).

F1F2F3

1.2

Uniform-norm continuity at $e$ for compactly supported functions. Let $f \in C_{c} (G)$ . By [F4] applied at $a_{0} = e$ there are a neighbourhood $V_{0}$ of $e$ and a compact $C \subseteq G$ containing the supports of $L_{a} f$ , $ρ_{a} f$ and $f$ for every $a \in V_{0}$ , with $∥ L_{a} f - f ∥_{\infty} \to 0$ and $∥ ρ_{a} f - f ∥_{\infty} \to 0$ as $a \to e$ . The scalar $Δ_{G} (a)^{1 / 2}$ tends to $1$ by [F3], and $∥ R_{a} f - f ∥_{\infty} \leq Δ_{G} (a)^{1 / 2} ∥ ρ_{a} f - f ∥_{\infty} + ∣ Δ_{G} (a)^{1 / 2} - 1 ∣ ∥ f ∥_{\infty} ⟶ 0.$ Since this scalar is positive, $supp R_{a} f = supp ρ_{a} f \subseteq C$ . Thus for every $ϵ > 0$ some neighbourhood $V \subseteq V_{0}$ makes both $∥ L_{a} f - f ∥_{\infty}$ and $∥ R_{a} f - f ∥_{\infty}$ less than $ϵ$ for all $a \in V$ .

F3F4

2.1

Compactly supported convergence in $L^{p}$ . With $f$ , $C$ and $V$ as in step 1.2, for $a \in V$ the difference $L_{a} f - f$ is supported in the compact set $C$ , so $∥ L_{a} f - f ∥_{1} \leq ∥ L_{a} f - f ∥_{\infty} μ (C)$ and $∥ L_{a} f - f ∥_{2} \leq ∥ L_{a} f - f ∥_{\infty} μ (C)^{1 / 2}$ , both tending to $0$ as $a \to e$ by the uniform bound of step 1.2, with $μ (C) < \infty$ by [F6]; the same estimates hold for $R_{a} f - f$ . Thus $L_{a} \to i d$ strongly on $C_{c} (G)$ for $p = 1, 2$ and $R_{a} \to i d$ strongly on $C_{c} (G)$ for $p = 2$ .

F6step 1.2

3.1

Left translations on $L^{p} (G)$ . Fix $f \in L^{p} (G)$ with $p \in {1, 2}$ and $ϵ > 0$ . By [F5] under the AC of [A1] choose $h \in C_{c} (G)$ with $∥ f - h ∥_{p} < ϵ / 3$ , and by step 2.1 choose a neighbourhood $V$ of $e$ with $∥ L_{a} h - h ∥_{p} < ϵ / 3$ for $a \in V$ . For $a \in V$ the isometry of step 1.1 gives $∥ L_{a} f - f ∥_{p} \leq ∥ L_{a} (f - h) ∥_{p} + ∥ L_{a} h - h ∥_{p} + ∥ h - f ∥_{p} < 3 \cdot ϵ / 3 = ϵ$ , so $L_{a} \to i d$ strongly at $e$ on $L^{p} (G)$ . For a general $g_{0} \in G$ , the definition $L_{a} f (x) = f (a^{- 1} x)$ gives $L_{a} L_{b} = L_{a b}$ , hence $L_{g} = L_{g_{0}} L_{g_{0}^{- 1} g}$ . Therefore $∥ L_{g} f - L_{g_{0}} f ∥_{p} = ∥ L_{g_{0}} (L_{g_{0}^{- 1} g} f - f) ∥_{p} = ∥ L_{g_{0}^{- 1} g} f - f ∥_{p},$ by the isometry of step 1.1. This is $< ϵ$ once $g_{0}^{- 1} g \in V$ , i.e. for $g$ in the neighbourhood $g_{0} V$ of $g_{0}$ .

A1F5step 1.1step 2.1

3.2

Right translations on all of $L^{2} (G)$ . Fix $f \in L^{2} (G)$ and $ϵ > 0$ , choose $h \in C_{c} (G)$ with $∥ f - h ∥_{2} < ϵ / 3$ and a neighbourhood $V$ of $e$ with $∥ R_{a} h - h ∥_{2} < ϵ / 3$ for $a \in V$ , by step 2.1 and [F5]. Since $R_{a}$ is isometric on $L^{2} (G)$ by step 1.1, $∥ R_{a} f - f ∥_{2} \leq ∥ R_{a} (f - h) ∥_{2} + ∥ R_{a} h - h ∥_{2} + ∥ h - f ∥_{2} < ϵ$ for $a \in V$ . For a general $g_{0}$ , $R_{g} = R_{g_{0}} R_{g_{0}^{- 1} g}$ because $Δ_{G} (g_{0})^{1 / 2} Δ_{G} (g_{0}^{- 1} g)^{1 / 2} = Δ_{G} (g)^{1 / 2}$ by multiplicativity [F3], so $∥ R_{g} f - R_{g_{0}} f ∥_{2} = ∥ R_{g_{0}} (R_{g_{0}^{- 1} g} f - f) ∥_{2} = ∥ R_{g_{0}^{- 1} g} f - f ∥_{2} < ϵ$ for $g \in g_{0} V$ , again by the isometry of step 1.1.

F3F5step 1.1step 2.1

4.1

Collecting the statements: step 3.1 gives strong continuity of $g \mapsto L_{g}$ on $L^{1} (G)$ and on $L^{2} (G)$ at every point, and step 3.2 gives strong continuity of $g \mapsto R_{g}$ on $L^{2} (G)$ at every point. ∎

step 3.1step 3.2

Remarks

Neighbourhoods, not sequences. Every convergence statement above is formulated with a neighbourhood $V$ of the relevant point, so it applies to a group of arbitrary cardinality and character; no sequential criterion ( $f$ is continuous at $c \in A$ if and only if $f (x_{k}) \to f (c)$ for every sequence in $A$ converging to $c$ , the converse direction costing countable choice formalises such a criterion only in first-countable spaces) is used.
Why $R_{g}$ carries the factor $Δ_{G} (g)^{1 / 2}$ . Without it the right translation would scale the $L^{2}$ norm by $Δ_{G} (g)^{- 1 / 2}$ and would fail to be isometric, hence could not be a unitary representation (Left and right regular unitary representations of an LCH group).
Choice cost. [A1] is inherited only through the density statement [F5]; the rest of the proof uses the fixed measure, its modular function and uniform continuity.

Depends on

Used by

Dependency tree · two levels

33 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

Lynn Loomis, An Introduction to Abstract Harmonic Analysis, §§30–31 (standard reference, not scraped)
Bekka, de la Harpe and Valette, Kazhdan’s Property (T), Appendix A §§A.3–A.4 (standard reference, not scraped)
Emmanuel Kowalski, Representation Theory of Groups, §§5.2–5.3 (standard reference, not scraped)