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Kernel inclusion implies weak containment

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group and let $π$ and $ρ$ be strongly continuous unitary representations whose extended representations of $C^{*} (G)$ (Nondegenerate representations of the full group C star algebra are unitary representations, The full (maximal) group C star algebra) satisfy $\ker ρ \subseteq \ker π$ . Then $π ≺ ρ$ (Weak containment of unitary representations).

Facts & Assumptions

Given: AC; an LCH group $G$ ; unitary representations $π, ρ$ with $\ker ρ \subseteq \ker π$ ; unit vectors $ξ \in H_{π}$ .

[F1]

$φ (a) : = ⟨ π (a) ξ, ξ ⟩$ is a state of $C^{*} (G)$ for every unit vector $ξ$ (States and positive functionals on a C star algebra); it vanishes on $\ker ρ \subseteq \ker π$ and therefore factors as $φ = \tilde{φ} \circ ρ$ with $\tilde{φ}$ a state of the C*-algebra $B : = ρ (C^{*} (G)) \subseteq B (K_{ρ})$ , because $C^{*} (G) / \ker ρ ≅ B$ isometrically (Quotients of C star algebras by closed two-sided ideals).

[F2]

Every state of $B$ is a weak-* limit of a net of convex combinations of normalized vector states: for suitable nets $θ_{i} = \sum_{j} λ_{i, j} ω_{η_{i, j}}$ , with $∥ η_{i, j} ∥ = 1$ , $λ_{i, j} \geq 0$ , $\sum_{j} λ_{i, j} = 1$ , one has $θ_{i} (b) \to \tilde{φ} (b)$ for every $b \in B$ (States of a concretely represented C star algebra are weak star limits of finite sums of vector states).

[F3]

Translation estimates: for a normalized coefficient $ψ (x) = ⟨ σ (x) η, η ⟩$ , $∥ η ∥ = 1$ , one has $∣ ψ (x h) - ψ (x) ∣ \leq (2 (1 - Re ψ (h)))^{1 / 2}$ (Translation estimates for continuous positive type functions, Continuous positive-type functions and normalization).

[F4]

For $f \in C_{c} (G)$ the integrated forms give $⟨ π (L_{g} f) ξ, ξ ⟩ = \int_{G} f (h) ⟨ π (g h) ξ, ξ ⟩ d h$ and likewise for $ρ$ and for vector functionals; the maps $g \mapsto L_{g} f$ are continuous in $L^{1} (G)$ with $∥ L_{g} f ∥_{C^{*}} \leq ∥ L_{g} f ∥_{1} = ∥ f ∥_{1}$ (The integrated form of a unitary representation, Strong continuity of left and modular right translations on L1 and L2, Integrated forms are contractive nondegenerate star representations of L one, The full (maximal) group C star algebra).

Proof

technique · direct

Given: AC, an LCH group $G$ , unitary representations $π, ρ$ with $\ker ρ \subseteq \ker π$ , a unit vector $ξ \in H_{π}$ , a compact set $Q \subseteq G$ and $ϵ > 0$ .

1.1F3F4

Let $k (g) : = ⟨ π (g) ξ, ξ ⟩$ and, for a convex combination $θ = \sum_{j} λ_{j} ω_{η_{j}}$ of normalized vector states of $B$ with associated coefficient $k_{θ} (g) : = \sum_{j} λ_{j} ⟨ ρ (g) η_{j}, η_{j} ⟩$ , and for $f \in C_{c} (G)$ with $\int_{G} f = 1$ and $f \geq 0$ , one has $∣ k (g) - ⟨ π (f) ξ, ξ ⟩_{g} ∣ \leq (2 (1 - Re φ (f)))^{1 / 2}$ and $∣ k_{θ} (g) - θ (L_{g} f) ∣ \leq (2 (1 - Re θ (f)))^{1 / 2}$ for every $g$ , where $⟨ π (f) ξ, ξ ⟩_{g} : = \int_{G} f (h) k (g h) d h$ and $θ (L_{g} f) = \int_{G} f (h) k_{θ} (g h) d h$ by [F4]. Indeed, $∣ k (g) - k (g h) ∣ \leq (2 (1 - Re k (h)))^{1 / 2}$ by [F3], and Cauchy–Schwarz for the probability measure $f d h$ gives $\int_{G} f (h) (2 (1 - Re k (h)))^{1 / 2} d h \leq (2 (1 - \int_{G} f (h) Re k (h) d h))^{1 / 2} = (2 (1 - Re φ (f)))^{1 / 2}$ ; the same computation applies to $k_{θ}$ , whose summands satisfy the same estimate by [F3] and Cauchy–Schwarz for the weights $λ_{j}$ .

1.2F1F2F4

Fix $f \in C_{c} (G)$ . The set ${L_{g} f : g \in Q}$ is compact in $L^{1} (G)$ by [F4], hence its image under the continuous map into $C^{*} (G)$ is compact; since $θ_{i} (ρ (b)) \to \tilde{φ} (ρ (b))$ for every $b \in C^{*} (G)$ by [F1] and [F2], a finite $δ$ -net argument gives $\sup_{g \in Q} ∣ θ_{i} (L_{g} f) - φ (L_{g} f) ∣ \to 0$ , where $φ (L_{g} f) = \tilde{φ} (ρ (L_{g} f))$ .

2.1F1step 1.1step 1.2

Consequently $k$ is a compact-uniform limit of the coefficients $k_{θ}$ : enlarging the given compact set $Q$ to $Q \cup {e}$ if necessary, and given $ϵ > 0$ , choose $f \in C_{c} (G)$ with $f \geq 0$ , $\int f = 1$ and support so small that $1 - Re k (h) < ϵ$ on it, so that $1 - Re φ (f) < ϵ$ ; eventually $1 - Re θ_{i} (f) < 2 ϵ$ by step 1.2 applied at $e$ , and then $\sup_{Q} ∣ k - k_{θ_{i}} ∣ \leq (2 ϵ)^{1 / 2} + \sup_{Q} ∣ θ_{i} (L_{g} f) - φ (L_{g} f) ∣ + (4 ϵ)^{1 / 2}$ , which is $< 4 \sqrt{ϵ}$ once $i$ is large: the first and third terms sum to $(\sqrt{2} + 2) \sqrt{ϵ} < 4 \sqrt{ϵ}$ , and the middle term tends to zero, by steps 1.1 and 1.2.

3.1step 2.1

Therefore every normalized diagonal coefficient of $π$ is a compact-uniform limit of finite sums of diagonal coefficients of $ρ$ ; for an arbitrary vector $ξ \neq 0$ the coefficient $c_{ξ, ξ} = ∥ ξ ∥^{2} c_{ξ / ∥ ξ ∥, ξ / ∥ ξ ∥}$ is a nonnegative multiple of a normalized one and the approximating sums scale by the same factor, so by [F1]–[F2] and the definition of weak containment $π ≺ ρ$ .

4.1givenF2∎

The Axiom of Choice is used for the geometric separation behind the vector-state approximation of step 2.1 and is inherited from the whole chain (The Axiom of Choice).

Depends on

Used by

Weak containment is equivalent to kernel inclusion Theorem

Dependency tree · two levels

61 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 and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019) (standard reference, not scraped)
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)