Haagerup property

Haagerup property

This property was originally formulated by Uffe Haagerup for locally compact groups (see Haagerup property). It was generalized to the context of locally compact quantum groups [DFSW13]. There is also a definition of a Haagerup property for von Neumann algebras [Jol02]. Here, we focus on the case of discrete quantum groups (note that every compact quantum groups has the Haagerup property).

Definition

Let $\Gamma$ be a discrete quantum group. The following equivalent statements provide a definition of $\Gamma$ having a Haagerup property [DFSW13]

There exists a mixing representation of $\Gamma$ which has almost invariant vectors.
There is a net of sates $(\mu_i)$ on $O(\hat\Gamma)$ such that…
$\hat\Gamma$ admits a symmetric proper generating functional
$\Gamma$ admits a proper real cocycle

A von Neumann algebra $M$ equipped with a faithful normal trace $\tau$ is said to have the Haagerup property if there exists a net of $\tau$ -preserving unital completely positive maps $\theta_i$ on $M$ such that each $\theta_i$ extends to a compact operator on $L^2(M,\tau)$ and $\theta_i\to\id_M$ in the point-ultraweak topology.

An unimodular discrete quantum group $\Gamma$ has the Haagerup property if and only if $L^\infty(\hat\Gamma)$ has the Haagerup property.

Results

Stability results

If $\Gamma$ has the Haagerup property, then any quantum subgroup of $\Gamma$ has the Haagerup property. [DFSW13]
Let $G$ be a CQG, $H\subset G$ a finite index subgroup (i.e. $C(G/H):=\{a\in C_{\rm u}(G)\mid (\id\otimes p)\Delta(a)=a\otimes 1\}$ , where $p$ is the surjection $C_{\rm u}(G)\to C_{\rm u}(H)$ , is finite dimensional). If $\hat H$ has the Haagerup property, then $\hat G$ has the Haagerup property. [FMP17]
Free product $\Gamma_1*\Gamma_2$ has the Haagerup property if and only if both $\Gamma_1$ and $\Gamma_2$ have the Haagerup property [DFSW13]

Examples

Any finite quantum group
Coxeter groups
$\mathbb{F}_n$ [Haa78], $\mathrm{SL}_2(\Z)$
$\hat O_N^+$ , $\hat U_N^+$ [Bra11]
$\hat O_F^+$ , $\hat U_F^+$ [DFY14]
$S_N^+$ , $H_N^{s+}$ [Lem15]

Relation with other properties

If $\Gamma=\hat G$ has the Haagerup property, then

$L^\infty(G)$ has the Haagerup approximation property [DFSW13]

Discrete quantum group $\Gamma=\hat G$ has the Haagerup property if

$\Gamma$ is amenable [DFSW13]

Discrete quantum group $\Gamma$ has the Haagerup property and property (T) if and only if $\Gamma$ is finite [DFSW13].

References

[DFSW13] Matthew Daws, Pierre Fima, Adam Skalski, Stuart White, 2013. The Haagerup property for locally compact quantum groups. Journal für die reine und angewandte Mathematik, 2016(711), pp.189–229.

[Jol02] Paul Jolissaint, 2002. Haagerup Approximation property for finite von Neumann algebras. Journal of Operator Theory, 48(3), pp.549–571.

[FMP17] Pierre Fima, Kunal Mukherjee and Issan Patri, 2017. On compact bicrossed products. Journal of Noncommutative Geometry, 11(4), pp.1521–1591.

[Haa78] Uffe Haagerup, 1978. An example of a non nuclearC*-algebra, which has the metric approximation property. Inventiones mathematicae, 50(3), pp.279–293.

[Bra11] Michael Brannan, 2011. Approximation properties for free orthogonal and free unitary quantum groups. Journal für die reine und angewandte Mathematik, 2012(672), pp.223–251.

[DFY14] Kenny De Commer, Amaury Freslon, Makoto Yamashita, 2014. CCAP for Universal Discrete Quantum Groups. Communications in Mathematical Physics, 331(2), pp.677–701.

[Lem15] François Lemeux, 2015. Haagerup approximation property for quantum reflection groups. Proceedings of the Americal Mathematical Society, 143, pp.2017–2031.

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