haagerup_property
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| ==== Stability 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)] | * 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)] | ||
| - | * If $\Gamma$ [[unimodularity|unimodular]] has Haagerup property, then also $\hat G\tilstar\Z$ has Haagerup property [(ref:Bra11)] | ||
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| * Any finite quantum group | * Any finite quantum group | ||
| + | * Coxeter groups | ||
| * $\mathbb{F}_n$ [(ref:Haa78)], $\mathrm{SL}_2(\Z)$ | * $\mathbb{F}_n$ [(ref:Haa78)], $\mathrm{SL}_2(\Z)$ | ||
| * $\hat O_N^+$, $\hat U_N^+$ [(ref:Bra11)] | * $\hat O_N^+$, $\hat U_N^+$ [(ref:Bra11)] | ||
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| ===== 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 |
| - | * $L^\infty(G)$ has the Haagerup approximation property | + | * $L^\infty(G)$ has the Haagerup approximation property [(ref:DFSW13)] |
| - | Discrete quantum group $\Gamma=\hat G$ has (T) if | + | Discrete quantum group $\Gamma=\hat G$ has the Haagerup property if |
| - | + | ||
| - | * $\Gamma$ is [[Amenability|amenable]] | + | |
| + | * $\Gamma$ 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. | ||
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| doi : 10.1007/s00220-014-2052-7 | doi : 10.1007/s00220-014-2052-7 | ||
| url : https://doi.org/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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