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The regular representations are unitary, strongly continuous, and the left one is faithful

Statement

Assume AC. Let $G$ be an LCH group with a fixed left Haar measure $μ$ , and let $λ, ρ$ be the left and right regular representations on $L^{2} (G)$ (Left and right regular unitary representations of an LCH group). Then $λ$ and $ρ$ are strongly continuous unitary representations of $G$ on the Hilbert space $L^{2} (G)$ , and $λ$ is faithful: $λ (g) = i d$ implies $g = e$ .

Facts & Assumptions

Given: An LCH group $G$ with fixed left Haar measure $μ$ , its modular function $Δ_{G}$ , the Hilbert space $L^{2} (G)$ , the maps $λ, ρ$ , and AC.

[F1]

$λ (g) ξ (x) = ξ (g^{- 1} x)$ and $ρ (g) ξ (x) = Δ_{G} (g)^{1 / 2} ξ (x g)$ define complex-linear isometries of $L^{2} (G)$ , with $λ (g)^{- 1} = λ (g^{- 1})$ , $ρ (g)^{- 1} = ρ (g^{- 1})$ , and group laws $λ (g h) = λ (g) λ (h)$ , $ρ (g h) = ρ (g) ρ (h)$ (Left and right regular unitary representations of an LCH group).

[F2]

Strong continuity: for $ξ \in L^{2} (G)$ and $ϵ > 0$ there is a neighbourhood $V$ of $e$ with $∥ λ (g) ξ - ξ ∥_{2} < ϵ$ and $∥ ρ (g) ξ - ξ ∥_{2} < ϵ$ for all $g \in V$ ; and at every point $g_{0}$ of $G$ (Strong continuity of left and modular right translations on L1 and L2).

[F3]

$L^{2} (G)$ is a Hilbert space with inner product $⟨ ξ, η ⟩ = \int_{G} ξ \overline{η} d μ$ , whose induced norm is $∥ \cdot ∥_{2}$ , and $C_{c} (G) = C_{c} (G; C)$ is dense in it (Hilbert space, Complex Haar L^p spaces and compactly supported functions, Completeness of the complex Haar L1 and L2 spaces and density of Cc, Compact support, $C_{c} (X)$ , and $C_{0} (X)$ ).

[F4]

In the terminology of the cited definition a representation is unitary when it takes values in the group of unitary operators, and strongly continuous when $g \mapsto π (g) ξ$ is continuous for every $ξ$ (Continuous and unitary representations).

[F5]

If $h \in L^{2} (G)$ vanishes a.e. then $h = 0$ : the set where a continuous representative is nonzero is open and thus has positive measure when nonempty, and a class vanishing a.e. has the zero class as its only continuous representative (Haar measure is positive on nonempty open sets and finite on compact sets).

[F6]

Under Dependent Choice, if $K$ is compact and contained in an open $U$ , there is $f \in C_{c} (G)$ with $1_{K} \leq f \leq 1_{U}$ ; AC implies Dependent Choice. Taking $K = {e}$ gives a nonzero $f$ which vanishes off $U$ (LCH Urysohn cutoff, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[A1]

AC is assumed, in the choice-function form of the cited definition; it is used in step 2.2 through the cutoff function of [F6] (The Axiom of Choice).

Proof

technique · direct

1.1

The group laws already hold by [F1]: $λ (g h) = λ (g) λ (h)$ , $ρ (g h) = ρ (g) ρ (h)$ and $λ (e) = ρ (e) = i d$ , and the inverses are $λ (g^{- 1})$ , $ρ (g^{- 1})$ .

1.2

Each $λ (g)$ and $ρ (g)$ is a bijective complex-linear isometry of $L^{2} (G)$ : it is complex-linear and isometric by [F1], and surjective because the displayed inverse is a two-sided inverse. In the Hilbert-space setting this makes each of them a unitary operator, in the sense recalled in [F4] and in the definition [F1].

F1F4

1.3

Strong continuity. By [F2] the maps $g \mapsto λ (g) ξ$ and $g \mapsto ρ (g) ξ$ are continuous at $e$ for every $ξ \in L^{2} (G)$ ; continuity at an arbitrary $g_{0} \in G$ follows because $∥ λ (g) ξ - λ (g_{0}) ξ ∥_{2} = ∥ λ (g_{0}^{- 1} g) ξ - ξ ∥_{2} \to 0$ as $g \to g_{0}$ and likewise for $ρ$ , using the group laws and isometry of [F1]. Hence $λ$ and $ρ$ are strongly continuous in the sense of [F4].

F1F2F4

2.1

Unitarity of the two homomorphisms. By steps 1.1 and 1.2, $λ$ and $ρ$ are group homomorphisms $G \to U (L^{2} (G))$ ; by step 1.3 they are strongly continuous. Thus both are strongly continuous unitary representations of $G$ on the Hilbert space $L^{2} (G)$ .

F4step 1.1step 1.2step 1.3

2.2

Faithfulness of $λ$ . Suppose $g \neq e$ and $λ (g) = i d$ . Since $G$ is Hausdorff, choose an open neighbourhood $W$ of $e$ with $g \notin W$ . The map $q (x, y) = x y^{- 1}$ is continuous and $q (e, e) = e$ , so there are open neighbourhoods $V_{1}, V_{2}$ of $e$ with $V_{1} V_{2}^{- 1} \subseteq W$ ; their intersection $V_{0} = V_{1} \cap V_{2}$ satisfies $V_{0} V_{0}^{- 1} \subseteq W$ . By local compactness choose a compact neighbourhood $C$ of $e$ and an open neighbourhood $O$ with $e \in O \subseteq C$ . The set $C$ is closed because $G$ is Hausdorff, so $U : = V_{0} \cap O$ is an open neighbourhood of $e$ with compact closure contained in $C$ . Since $U U^{- 1} \subseteq W$ and $g \notin W$ , one has $U \cap g U = \emptyset$ . By [F6], under the Dependent Choice derived from [A1], choose $f \in C_{c} (G)$ with $1_{{e}} \leq f \leq 1_{U}$ . Then $f (e) \geq 1$ , $f$ vanishes off $U$ , and $f \in L^{2} (G)$ by [F3]. The class $λ (g) f - f$ is zero since $λ (g) = i d$ . But ${x : f (x) > 0}$ is a nonempty open set; for each such $x$ we have $x \in U$ and $x \notin g U$ , so $g^{- 1} x \notin U$ and $(λ (g) f) (x) = f (g^{- 1} x) = 0$ . Thus $λ (g) f - f = - f$ on this nonempty open set, so it is not the zero class by [F5], a contradiction. Hence $λ$ is faithful.

A1F3F5F6step 1.1

3.1

Combining the steps: $λ$ and $ρ$ are strongly continuous unitary representations by step 2.1, and $λ$ is faithful by step 2.2. ∎

step 2.1step 2.2

Remarks

Why the left representation is the faithful one. Both $λ$ and $ρ$ are faithful as well whenever $G$ is such that $ρ (g) = i d$ forces $g = e$ ; the statement records faithfulness only for $λ$ , because that is the case the proof above establishes directly using left translates.
Choice cost. [A1] is used only in step 2.2, for the cutoff function supported in a neighbourhood disjoint from its translate; the unitarity and strong continuity statements use only the fixed measure and its modular function.

Depends on

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Sources

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)