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Haar averaging projects contractively onto the bounded intertwiners

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $K$ be a compact Hausdorff group with normalized Haar probability measure $μ$ (Normalized Haar probability on a compact group), and let $π : K \to U (H)$ and $σ : K \to U (J)$ be strongly continuous unitary representations of $K$ on complex Hilbert spaces $H$ and $J$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Hilbert space). Write ${Hom}_{K} (H, J) : = {T \in B (H, J) : T π (g) = σ (g) T for every g \in K}$ for the space of bounded intertwiners and let $A : B (H, J) \to B (H, J)$ be the Haar averaging operator of Haar averaging of bounded operators as a weak operator integral, so that $⟨ A (T) v, w ⟩ = \int_{K} ⟨ σ (k) T π (k)^{- 1} v, w ⟩ d μ (k)$ for all $T \in B (H, J)$ , $v \in H$ , $w \in J$ . Then:

$A$ is idempotent, $A \circ A = A$ , and its range is exactly ${Hom}_{K} (H, J)$ : $A (T)$ intertwines for every bounded $T$ , and $A (T) = T$ for every bounded intertwiner $T$ ;
$A$ is a contraction, $∥ A (T) ∥ \leq ∥ T ∥$ for every $T \in B (H, J)$ , hence $∥ A ∥ \leq 1$ ;
$∥ A ∥ = 1$ if ${Hom}_{K} (H, J) \neq {0}$ , and $∥ A ∥ = 0$ if ${Hom}_{K} (H, J) = {0}$ .

Facts & Assumptions

Given: AC, a compact Hausdorff group $K$ with normalized Haar probability $μ$ , strongly continuous unitary representations $π$ on $H$ and $σ$ on $J$ , the averaging map $A$ of the definition item, and a bounded linear operator $T \in B (H, J)$ .

[F1]

The operator $A (T)$ is well defined by the weak operator integral: for all $v \in H$ and $w \in J$ the displayed pairing formula holds; the integrand is continuous on $K$ , hence bounded and integrable against $μ$ ; $A$ is linear in $T$ ; and $∥ A (T) ∥ \leq ∥ T ∥$ (Haar averaging of bounded operators as a weak operator integral, A bounded linear operator between normed spaces). The definition uses Countable Choice, which follows from AC, through the Riesz representation theorem for $J$ (Riesz representation for Hilbert spaces, Real and complex inner-product spaces and their induced length, AC supplies the countable and dependent choices used in Banach integration).

[F2]

The normalized Haar probability is left invariant and $μ (K) = 1$ ; for $h \in K$ the left translation $T_{h} (k) : = h k$ is a measurable self-map with $μ (T_{h}^{- 1} E) = μ (h^{- 1} E) = μ (E)$ , hence measure preserving, so $\int_{K} f (h k) d μ (k) = \int_{K} f (k) d μ (k)$ for every integrable $f$ (Normalized Haar probability on a compact group, Measure-preserving transformations and systems, Integral invariance under measure-preserving maps, Measure spaces).

[F3]

$π$ and $σ$ are group homomorphisms into the unitary groups, so $π (e) = {i d}_{H}$ , $π (h^{- 1} k)^{- 1} = π (k)^{- 1} π (h)$ , $σ (h k) = σ (h) σ (k)$ , and $σ (h)^{- 1} = σ (h^{- 1})$ ; each $σ (h)$ is unitary with $⟨ σ (h) x, y ⟩ = ⟨ x, σ (h)^{- 1} y ⟩$ . A bounded operator $T : H \to J$ is an intertwiner, written $T \in {Hom}_{K} (H, J)$ , exactly when $T π (g) = σ (g) T$ for every $g \in K$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[F4]

For bounded operators $∥ B v ∥ \leq ∥ B ∥ ∥ v ∥$ for every vector $v$ , the operator norm is the supremum of $∥ B v ∥$ over the closed unit ball (also when the domain is zero), and $∥ B ∥ = 0$ exactly when $B = 0$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces).

Proof

technique · direct

1.1F1F2F3

Let $T \in B (H, J)$ , $h \in K$ , $v \in H$ and $w \in J$ . Using unitarity of $σ (h)$ and the pairing formula of [F1], $⟨ σ (h) A (T) v, w ⟩ = ⟨ A (T) v, σ (h)^{- 1} w ⟩ = \int_{K} ⟨ σ (k) T π (k)^{- 1} v, σ (h)^{- 1} w ⟩ d μ (k) = \int_{K} ⟨ σ (h) σ (k) T π (k)^{- 1} v, w ⟩ d μ (k) = \int_{K} ⟨ σ (h k) T π (k)^{- 1} v, w ⟩ d μ (k)$ by the homomorphism property of [F3]. Substituting $k = h^{- 1} k^{'}$ and using left invariance of $μ$ as recorded in [F2], this equals $\int_{K} ⟨ σ (k^{'}) T π (h^{- 1} k^{'})^{- 1} v, w ⟩ d μ (k^{'}) = \int_{K} ⟨ σ (k^{'}) T π (k^{'})^{- 1} π (h) v, w ⟩ d μ (k^{'}) = ⟨ A (T) π (h) v, w ⟩$ , by [F3] and the pairing formula of [F1] again. As $v, w$ are arbitrary, $σ (h) A (T) = A (T) π (h)$ , and as $h$ is arbitrary, $A (T)$ is an intertwiner.

1.2F1F3

Let $T \in {Hom}_{K} (H, J)$ and $k \in K$ . Then $σ (k) T π (k)^{- 1} = σ (k) σ (k)^{- 1} T = T$ by the intertwining relation and the homomorphism properties of [F3]. Hence the integrand of the pairing formula is the constant $k \mapsto ⟨ T v, w ⟩$ , whose integral against the probability measure $μ$ is again $⟨ T v, w ⟩$ ; therefore $⟨ A (T) v, w ⟩ = ⟨ T v, w ⟩$ for all $v, w$ and $A (T) = T$ .

1.3F1F4

For every $T \in B (H, J)$ the definition of $A$ gives $∥ A (T) ∥ \leq ∥ T ∥$ by [F1], so the operator norm of the linear map $A$ satisfies $∥ A ∥ \leq 1$ by the supremum description in [F4].

2.1F1step 1.1step 1.2

Let $T \in B (H, J)$ . By step 1.1 the operator $A (T)$ lies in ${Hom}_{K} (H, J)$ , and step 1.2 applied to that intertwiner gives $A (A (T)) = A (T)$ . Hence $A \circ A = A$ , so $A$ is idempotent, and $range (A) \subseteq {Hom}_{K} (H, J)$ ; conversely every $S \in {Hom}_{K} (H, J)$ satisfies $S = A (S)$ by step 1.2, so ${Hom}_{K} (H, J) \subseteq range (A)$ . Thus the range of $A$ is exactly ${Hom}_{K} (H, J)$ .

3.1F4step 1.2step 1.3step 2.1∎

If ${Hom}_{K} (H, J) \neq {0}$ , choose a nonzero intertwiner $S$ . Then $S = A (S)$ by step 1.2, so $∥ S ∥ = ∥ A (S) ∥ \leq ∥ A ∥ ∥ S ∥$ by [F4], and since $∥ S ∥ \neq 0$ this gives $∥ A ∥ \geq 1$ ; with step 1.3, $∥ A ∥ = 1$ . If instead ${Hom}_{K} (H, J) = {0}$ , then $range (A) = {0}$ by step 2.1, so $A (T) = 0$ for every $T$ and $∥ A ∥ = 0$ by [F4]. Together with the contraction bound this proves the precise norm statement.

Depends on

Used by

Schur orthogonality for general compact groups Theorem

Cited to discharge well-definedness by Haar averaging of bounded operators as a weak operator integral.

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Sources

David Vogan, Review of Harmonic Analysis on Compact Groups, §§2.1–2.16 (standard reference, not scraped)
Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, §§5.2–5.6 (standard reference, not scraped)