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Bounds and two-sided uniform continuity of unitary coefficients

Statement

Let $G$ be a topological group and let $π : G \to U (H)$ be a strongly continuous unitary representation on a complex Hilbert space, with the inner product linear in its first argument. For $ξ, η \in H$ , define $c_{ξ, η} (g) = ⟨ π (g) ξ, η ⟩$ . Then

$∣ c_{ξ, η} (g) ∣ \leq ∥ ξ ∥ ∥ η ∥ (g \in G),$

and $c_{ξ, η} : G \to C$ is uniformly continuous for each of the left and right uniformities defined by $L_{U} = {(x, y) : x^{- 1} y \in U}$ and $R_{U} = {(x, y) : y x^{- 1} \in U}$ , with the usual metric uniformity on $C$ .

Facts & Assumptions

[A1]

The coefficient is $c_{ξ, η} (g) = ⟨ π (g) ξ, η ⟩$ (Matrix coefficient of a unitary representation).

[A2]

The representation is a group homomorphism $π : G \to U (H)$ , where $U (H)$ consists of bijective complex-linear isometries (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[A3]

Every orbit map $g \mapsto π (g) v$ is norm-continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[A4]

The pairing is linear in its first argument, conjugate-linear in its second, conjugate symmetric, and induces the norm by $⟨ v, v ⟩ = ∥ v ∥^{2}$ (Real and complex inner-product spaces and their induced length).

[A5]

For all $u, v \in H$ , $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A6]

The left and right group uniformities have basic entourages $L_{U} = {(x, y) : x^{- 1} y \in U}$ and $R_{U} = {(x, y) : y x^{- 1} \in U}$ for identity neighbourhoods $U$ (The left and right uniformities of a topological group).

[A7]

Inversion $h \mapsto h^{- 1}$ is continuous on $G$ (Topological group: multiplication and inversion are continuous).

[A8]

The usual complex metric is $d_{C} (z, w) = ∣ z - w ∣$ (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane).

[A9]

For a metric space, the usual metric uniformity has basic entourages ${(z, w) : d_{C} (z, w) < ε}$ , $ε > 0$ (A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated).

[A10]

A map between uniform spaces is uniformly continuous when each target entourage contains the image of some source entourage (Uniformly continuous map between uniform spaces).

Proof

technique · direct

Given: $G, π, H, ξ, η$ as in the statement. All inner products below are linear in the first variable.

1.1A2A4algebra

Every complex-linear norm isometry $U : H \to H$ preserves the inner product. For $z = ⟨ u, v ⟩$ , expansion using [A4] gives $∥ u + v ∥^{2} - ∥ u - v ∥^{2} = 4 Re z$ and $∥ u + i v ∥^{2} - ∥ u - i v ∥^{2} = 4 Im z$ . Since $U$ is complex-linear and preserves norms, these identities give equality of the real and imaginary parts of $⟨ U u, U v ⟩$ and $⟨ u, v ⟩$ . In particular, for every $h \in G$ , $⟨ π (h) u, v ⟩ = ⟨ u, π (h^{- 1}) v ⟩,$ because $π (h^{- 1})$ is the inverse of $π (h)$ .

1.2A1A2A5

Cauchy–Schwarz and the isometry property give, for every $g \in G$ , $∣ c_{ξ, η} (g) ∣ = ∣ ⟨ π (g) ξ, η ⟩ ∣ \leq ∥ π (g) ξ ∥ ∥ η ∥ = ∥ ξ ∥ ∥ η ∥ .$ If either vector is zero, this also says directly that the coefficient is identically zero.

1.3A1A2A3A4A5A6

Fix $ε > 0$ . By orbit continuity at $e$ , choose an identity neighbourhood $U$ such that $∥ π (h) ξ - ξ ∥ < ε / (1 + ∥ η ∥)$ for every $h \in U$ . If $(x, y) \in L_{U}$ , then $h = x^{- 1} y \in U$ and $y = x h$ . The homomorphism law, linearity in the first argument, and [A5] give $∣ c_{ξ, η} (y) - c_{ξ, η} (x) ∣ = ∣ ⟨ π (x) (π (h) ξ - ξ), η ⟩ ∣ \leq ∥ π (h) ξ - ξ ∥ ∥ η ∥ < ε .$ The same $U$ works for every $x$ , so this is uniform continuity for the left uniformity $L_{U}$ .

2.1A1A2A3A4A5A6A7step 1.1

Again fix $ε > 0$ . Orbit continuity for $η$ gives an identity neighbourhood $V$ such that $∥ π (k) η - η ∥ < ε / (1 + ∥ ξ ∥)$ for every $k \in V$ . By [A7], shrink to an identity neighbourhood $U$ such that $h \in U$ implies $h^{- 1} \in V$ . If $(x, y) \in R_{U}$ , put $h = y x^{- 1} \in U$ , so $y = h x$ . By step 1.1 and [A5], $∣ c_{ξ, η} (y) - c_{ξ, η} (x) ∣ = ∣ ⟨ π (x) ξ, π (h^{- 1}) η - η ⟩ ∣ \leq ∥ ξ ∥ ∥ π (h^{- 1}) η - η ∥ < ε .$ This $U$ works for every $x$ , proving uniform continuity for the right uniformity $R_{U}$ .

3.1A8A9A10step 1.2step 1.3step 2.1∎

Steps 1.3 and 2.1 give the entourage condition in [A9] for every metric entourage of $C$ , hence both asserted uniform continuities by [A10]. If $H = {0}$ , or if either coefficient vector is zero, the function is identically zero and all conclusions hold. No commutativity, local compactness, Haar measure, or choice is used.

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Sources

Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1, Appendix A printed pp. 305–306, and the left/right uniform-continuity convention in §A.3 printed pp. 318–319 (standard reference, not scraped)
Neeb, An Introduction to Unitary Representations of Lie Groups, §§3.4, 4.2 and 5.3 (standard reference, not scraped)