Groups with the Haagerup Property: Gromov’s a-T-menability
By P. Cherix, M. Cowling, Paul Jolissaint, P. Julg, Alain Valette. 2001
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View Paper: https://www.semanticscholar.org/paper/a238fcb6cb069aec9a81ab7e404c21a58d0f57fc
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- Non-Hausdorff groupoids, proper actions and $K$-theory (2004, background): “ arbitrary section x→ξx such that kgξs(g) −ξr(g)ktends to +∞when g→∞in G. Then (A1) implies (A2) (using properness of the second projection X′ ⋊G→ G), and (A2) implies (AT) (like in [4, 26], see also [8]). In Example 2.3 with H= {1}, Gsatisfies (A2) if and only if Γ is amenable, and Gsatisfies (AT) if”
- Examples of groups that are measure equivalent to the free group (2005, background): “n considered as opposite to amenability. More recently, the general opinionswitched, afterM. Gromovto opposeKazhdanpropertyT with the greaterclassofgroupswith Haagerup property (= a-T-menability, see [CCJJV01]). P ME 11 (Kazhdan Property T) Kazhdan property T is a ME invariant [Fur99a]. Lattices of ”
- Some geometric groups with Rapid Decay (2003, methodology): “0) cube complex using the work of [39] combined with the work of Higson and Kasparov [20] on a-(T)-menable groups satisfying the Baum-Connes conjecture (see also the exposition of P. Julg in [28] and [8] for a-(T)-menable groups). However, we mention the following consequence which follows immediate”
- HAAGERUP APPROXIMATION PROPERTY FOR FINITE VON NEUMANN ALGEBRAS (2002, background): “The first result of this section is already known for semidirect product groups H oα Γ whereH has the Haagerup property and Γ is amenable (see [7], Chapter 6); moreover, it follows directly from Proposition 4.”
- On property (T) for pairs of topological groups (2005, background): “4 in [4], we say that H is co-Følner in G if there exists a G-invariant state on L∞(G/H) (equivalently, if the homogeneous space G/H is amenable in the sense of Eymard [6]).”
- Kazhdan and Haagerup properties from the median viewpoint (2007, unknown): “(See [29,36,1,28,24].”
- Smooth Fourier multipliers on group von Neumann algebras (2010, background): “k), we consider the linear map B: H → L2(Ω,µ) given by B(h) = P khh,ekiH γk. Let Σ stand for smallest σ-algebramaking all the B(h)’s measurable. Then the well-knowngaussian measure space construction [3] tells us that, for every real unitary σ∈ O(H), we can construct a measure preserving automorphis”
- New C*‐completions of discrete groups and related spaces (2012, background): “he previous lemma implies the GNS representations associated to the ϕn are D-representations, concluding the proof. It has been open for some time whether the Haagerup property (≡ a-T-menability, see [6]) admits a C∗-algebraic characterization. The previous theorem easily implies such a characteriza”
- A class of groups for which every action is W$^*$-superrigid (2010, background): “ Γ has the Haagerup property if and only if there exists a sequence of positive definite functions ϕn : Γ → Ctending to 1 pointwise and such that for every n the function ϕn belongs to c0(Γ) (see e.g. [CCJJV]). 2 • If M has property (H) relative to P ⊂ M, there exists a sequence of normal subunital s”
- Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers (2014, background): “ (ej)j≥1, we consider the linear map B : H → L2(Ω,µ) given by B(ej) = γj. Let Σ stand for smallest σ-algebra making all the B(h)’s measurable. Then, the well known gaussian measure space construction [12] tells us that, for every real unitary αin O(H), we can construct a measure preserving automorph”