Math 2.0

Topological groups, Lie groups Definition

Haagerup property

Builds on Locally compact groupUnitary representationHilbert spaceCompact space

Definition

Let GG be a second countable locally compact group. A unitary representation π\pi of GG on a Hilbert space ℋ\mathcal{H} is mixing if ⟨π(g)ξ,η⟩→0\ip{\pi(g)\xi}{\eta} \to 0 as g→∞g \to \infty, for all ξ,η∈ℋ\xi, \eta \in \mathcal{H}. Here g→∞g \to \infty means that gg leaves every compact subset of GG. The representation π\pi almost has invariant vectors if the following holds. For every compact Q⊆GQ \subseteq G and ε>0\varepsilon > 0, there is a unit vector ξ\xi with ∥π(g)ξ−ξ∥<ε\norm{\pi(g)\xi - \xi} < \varepsilon for all g∈Qg \in Q. The group GG has the Haagerup property if it has a mixing unitary representation that almost has invariant vectors.

Remarks

Example: G=ℤG = \Z and the regular representation on ℓ2(ℤ)\ell^2(\Z). For finitely supported ξ\xi and η\eta, the coefficient ⟨π(n)ξ,η⟩\ip{\pi(n)\xi}{\eta} is 00 once ∣n∣\abs{n} is large, and by density the representation is mixing. The unit vectors N−1/2𝟏{0,…,N−1}N^{-1/2} \mathbf{1}_{\set{0, \ldots, N-1}} are almost invariant for large NN.

Non-example: SL3(ℤ)\mathrm{SL}_3(\Z) does not have the Haagerup property. It has property (T), so a representation that almost has invariant vectors has a nonzero invariant vector ξ\xi. Then ⟨π(g)ξ,ξ⟩=∥ξ∥2\ip{\pi(g)\xi}{\xi} = \norm{\xi}^2 for all gg, which does not tend to 00 on the infinite group.

Equivalently, GG has a proper continuous conditionally negative definite function ψ ⁣:G→[0,∞)\psi \colon G \to [0, \infty). Another equivalent form: GG has a continuous affine isometric action on a Hilbert space that is metrically proper. Amenable groups, free groups and SL2(ℝ)\mathrm{SL}_2(\R) have the Haagerup property.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Compact space, concept, no evidence levelCompact spaceconceptHilbert space, concept, no evidence levelHilbert spaceconceptLocally compact group, concept, no evidence levelLocally compact groupconceptUnitary representation, concept, no evidence levelUnitary representationconceptHaagerup property, definition, no evidence levelHaagerup propertydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Compact space concept
  2. Hilbert space concept
  3. Locally compact group concept
  4. Unitary representation concept
  5. Haagerup property definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Haagerup_property
About this page
TagEHGRPCTM3J
URIm2://EHGRPCTM3J
Kinddefinition
WritingW0 stub
MSC22D55, 20F65
DefinesHaagerup property
UsesLocally compact group, Unitary representation, Hilbert space, Compact space
Changed2 November 2026

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