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Weak containment is equivalent to kernel inclusion

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group and let $π$ and $ρ$ be strongly continuous unitary representations of $G$ , extended to nondegenerate star-representations of the full group C*-algebra $C^{*} (G)$ (Nondegenerate representations of the full group C star algebra are unitary representations, The full (maximal) group C star algebra). Write $\ker π : = {a \in C^{*} (G) : π (a) = 0}$ and similarly for $ρ$ . Then the following are equivalent:

$π ≺ ρ$ (Weak containment of unitary representations);
$\ker ρ \subseteq \ker π$ ;
$∥ π (a) ∥ \leq ∥ ρ (a) ∥$ for every $a \in C^{*} (G)$ .

In particular, for irreducible $π$ and $ρ$ one has $π \sim ρ$ if and only if $\ker π = \ker ρ$ ; consequently the kernel map $κ : \hat{G} \to Prim (C^{*} (G))$ , $κ ([π]) = \ker π$ , of The primitive ideal space of a group C star algebra is well defined on unitary equivalence classes and its fibres are exactly the weak equivalence classes.

Facts & Assumptions

Given: AC; an LCH group $G$ ; strongly continuous unitary representations $π, ρ$ of $G$ with their extensions to $C^{*} (G)$ ; $A = C^{*} (G)$ .

[F1]

If $π ≺ ρ$ then $∥ π (a) ∥ \leq ∥ ρ (a) ∥$ for every $a \in A$ ; in particular $\ker ρ \subseteq \ker π$ (Weak containment implies kernel inclusion).

[F2]

If $\ker ρ \subseteq \ker π$ then $π ≺ ρ$ (Kernel inclusion implies weak containment).

[F3]

Unitary representations of $G$ correspond bijectively, up to unitary equivalence, to nondegenerate star-representations of $A$ , and irreducibility is preserved on both sides; unitarily equivalent representations have equal kernels, which are closed two-sided ideals (Nondegenerate representations of the full group C star algebra are unitary representations, The full (maximal) group C star algebra).

[F4]

$\hat{G}$ is the set of unitary equivalence classes of irreducible strongly continuous unitary representations; primitive ideals and the kernel map $κ ([π]) = \ker π$ are as in The primitive ideal space of a group C star algebra (The unitary dual of a locally compact group).

Proof

technique · direct

Given: AC, an LCH group $G$ , unitary representations $π, ρ$ and their extensions to $A = C^{*} (G)$ .

1.1F1

Condition 1 implies conditions 2 and 3: by [F1], $π ≺ ρ$ gives $∥ π (a) ∥ \leq ∥ ρ (a) ∥$ for all $a \in A$ , and $a \in \ker ρ$ then gives $∥ π (a) ∥ \leq 0$ , that is $a \in \ker π$ .

1.2F2

Condition 2 implies condition 1: this is exactly [F2].

2.1step 1.1step 1.2

Condition 3 implies condition 2: if $∥ π (a) ∥ \leq ∥ ρ (a) ∥$ for every $a$ and $a \in \ker ρ$ , then $∥ π (a) ∥ \leq ∥ ρ (a) ∥ = 0$ , so $a \in \ker π$ . Together with steps 1.1 and 1.2 this proves that 1, 2 and 3 are equivalent.

3.1step 2.1

For irreducible $π, ρ$ the equivalence specializes: $π \sim ρ$ means $π ≺ ρ$ and $ρ ≺ π$ , which by step 2.1 is equivalent to $\ker ρ \subseteq \ker π$ and $\ker π \subseteq \ker ρ$ , that is $\ker π = \ker ρ$ .

4.1F3F4step 3.1

The kernel map $κ$ is well defined and has the weak equivalence classes as fibres. If $[π] = [ρ]$ in $\hat{G}$ , the representations are unitarily equivalent, hence have equal kernels by [F3], so $κ ([π])$ does not depend on the chosen representative; the class is irreducible, so $κ ([π]) = \ker π$ is a closed two-sided ideal that is the kernel of an irreducible nondegenerate star-representation of $A$ , hence a primitive ideal, and $κ$ maps into $Prim (C^{*} (G))$ by [F4]. Two classes have the same image exactly when $\ker π = \ker ρ$ , which by step 3.1 is exactly $π \sim ρ$ .

5.1givenF1F2F3∎

The Axiom of Choice is inherited from the two implication lemmas and from the representation correspondence; the bookkeeping of conditions and fibres adds no choice (The Axiom of Choice).

Depends on

Used by

Dependency tree · two levels

34 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)