====== Haagerup property ====== This property was originally formulated by Uffe Haagerup for locally compact groups (see [[wp>Haagerup property]]). It was generalized to the context of locally compact quantum groups [(ref:DFSW13)]. There is also a definition of a Haagerup property for von Neumann algebras [(ref: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** [(ref: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 [[unimodularity|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. [(ref: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. [(ref: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 [(ref:DFSW13)] ==== Examples ==== * Any finite quantum group * Coxeter groups * $\mathbb{F}_n$ [(ref:Haa78)], $\mathrm{SL}_2(\Z)$ * $\hat O_N^+$, $\hat U_N^+$ [(ref:Bra11)] * $\hat O_F^+$, $\hat U_F^+$ [(ref:DFY14)] * $S_N^+$, $H_N^{s+}$ [(ref:Lem15)] ===== Relation with other properties ===== If $\Gamma=\hat G$ has the Haagerup property, then * $L^\infty(G)$ has the Haagerup approximation property [(ref:DFSW13)] Discrete quantum group $\Gamma=\hat G$ has the Haagerup property if * $\Gamma$ is [[Amenability|amenable]] [(ref:DFSW13)] Discrete quantum group $\Gamma$ has the Haagerup property and [[kazhdan_property|property (T)]] if and only if $\Gamma$ is [[compact_quantum_group#Finite quantum groups|finite]] [(ref:DFSW13)]. ===== Further reading ===== * Michael Brannan. //[[https://arxiv.org/abs/1605.01770|Approximation properties for locally compact quantum groups]]//, 2016. * Nathanial P. Brown and Narutaka Ozawa, //C*-algebras and Finite-Dimensional Approximations//, [[https://bookstore.ams.org/gsm-88|American Mathematical Society]], 2008. ===== References ===== [(ref:Bra11>> author : Michael Brannan title : Approximation properties for free orthogonal and free unitary quantum groups journal : Journal für die reine und angewandte Mathematik volume : 2012 number : 672 pages : 223–251 year : 2011 doi : 10.1515/CRELLE.2011.166 url : https://doi.org/10.1515/CRELLE.2011.166 )] [(ref:DFSW13>> author : Matthew Daws, Pierre Fima, Adam Skalski, Stuart White title : The Haagerup property for locally compact quantum groups journal : Journal für die reine und angewandte Mathematik volume : 2016 number : 711 pages : 189–229 year : 2013 doi : 10.1515/crelle-2013-0113 url : https://doi.org/10.1515/crelle-2013-0113 )] [(ref:Jol02>> url : http://www.jstor.org/stable/24715585 author : Paul Jolissaint journal : Journal of Operator Theory number : 3 pages : 549–571 title : Haagerup Approximation property for finite von Neumann algebras volume : 48 year : 2002 )] [(ref:Haa78>> author : Uffe Haagerup title : An example of a non nuclearC*-algebra, which has the metric approximation property journal : Inventiones mathematicae year : 1978 volume : 50 number : 3 pages : 279–293 doi : 10.1007/BF01410082 url : https://doi.org/10.1007/BF0141008 )] [(ref:Fre13>> title : Examples of weakly amenable discrete quantum groups journal : Journal of Functional Analysis volume : 265 number : 9 pages : 2164--2187 year : 2013 doi : 10.1016/j.jfa.2013.05.037 url : http://www.sciencedirect.com/science/article/pii/S0022123613002127 author : Amaury Freslon )] [(ref:Lem15>> title : Haagerup approximation property for quantum reflection groups journal : Proceedings of the Americal Mathematical Society volume : 143 pages : 2017–2031 year : 2015 doi : 10.1090/S0002-9939-2015-12402-1 url : http://dx.doi.org/10.1090/S0002-9939-2015-12402-1 author : François Lemeux )] [(ref:DFY14>> author : Kenny De Commer, Amaury Freslon, Makoto Yamashita title : CCAP for Universal Discrete Quantum Groups journal : Communications in Mathematical Physics year : 2014 volume : 331 number : 2 pages : 677–701 doi : 10.1007/s00220-014-2052-7 url : https://doi.org/10.1007/s00220-014-2052-7 )] [(ref:FMP17>> title : On compact bicrossed products journal : Journal of Noncommutative Geometry volume : 11 number : 4 pages : 1521--1591 year : 2017 doi : 10.4171/JNCG/11-4-10 url : http://dx.doi.org/10.4171/JNCG/11-4-10 author : Pierre Fima, Kunal Mukherjee and Issan Patri )] ~~REFNOTES ref ~~