haagerup_property
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| 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. | 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. | + | An [[unimodularity|unimodular]] discrete quantum group $\Gamma$ has the Haagerup property if and only if $L^\infty(\hat\Gamma)$ has the Haagerup property. |
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| ==== Stability results ==== | ==== Stability results ==== | ||
| - | * If $\hat G$ unimodular has Haagerup property, then also $\widehat{G\tilstar\Z}$ has Haagerup property [(ref:Bra11)] | + | * 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 ==== | ==== Examples ==== | ||
| + | * Any finite quantum group | ||
| + | * Coxeter groups | ||
| + | * $\mathbb{F}_n$ [(ref:Haa78)], $\mathrm{SL}_2(\Z)$ | ||
| * $\hat O_N^+$, $\hat U_N^+$ [(ref:Bra11)] | * $\hat O_N^+$, $\hat U_N^+$ [(ref:Bra11)] | ||
| + | * $\hat O_F^+$, $\hat U_F^+$ [(ref:DFY14)] | ||
| + | * $S_N^+$, $H_N^{s+}$ [(ref:Lem15)] | ||
| Line 34: | Line 41: | ||
| ===== Relation with other properties ===== | ===== Relation with other properties ===== | ||
| - | If $\Gamma=\hat G$ has Haagerup property, then | + | If $\Gamma=\hat G$ has the Haagerup property, then |
| - | ... | + | |
| - | Discrete quantum group $\Gamma=\hat G$ has (T) if | + | * $L^\infty(G)$ has the Haagerup approximation property [(ref:DFSW13)] |
| - | * $\Gamma$ is [[Amenability|amenable]] | + | 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 ===== | ===== 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. | * Nathanial P. Brown and Narutaka Ozawa, //C*-algebras and Finite-Dimensional Approximations//, [[https://bookstore.ams.org/gsm-88|American Mathematical Society]], 2008. | ||
| Line 82: | Line 91: | ||
| volume : 48 | volume : 48 | ||
| year : 2002 | 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 ~~ | ~~REFNOTES ref ~~ | ||
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