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--- haagerup_property [2019/09/13 13:49]
d.gromada
+++ haagerup_property [2021/11/23 11:56] (current)
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 ====== 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:DFSW16)]. 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).
+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:DFSW16)]
+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.
@@ Line 13: / Line 13: @@
-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 u.c.p. 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 [[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)]
-An unimodular discrete quantum group $\Gamma$ has the Haagerup property if and only if $L^\infty(\hat\Gamma)$ has the Haagerup property.
 ===== 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 =====
+  * 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.
@@ Line 34: / Line 58: @@
 ===== References =====
-[(ref:DFSW16>>
+[(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
@@ Line 41: / Line 77: @@
 number    : 711
 pages     : 189–229
-year      : 2016
+year      : 2013
 doi       : 10.1515/crelle-2013-0113
 url       : https://doi.org/10.1515/crelle-2013-0113
@@ Line 55: / Line 91: @@
 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 ~~

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