kazhdan_property
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| ====== Kazhdan property (T) ====== | ====== Kazhdan property (T) ====== | ||
| + | |||
| + | This property was originally formulated by David Kazhdan for locally compact groups (see [[wp>Kazhdan's_property_(T)|Kazhdan's property (T)]]). This article is about its generalization to the quantum group setting. | ||
| ===== Definition ===== | ===== Definition ===== | ||
| Line 5: | Line 7: | ||
| The definition of Kazhdan property (T) for discrete quantum groups was formulated in [(ref:Fim10)]. | The definition of Kazhdan property (T) for discrete quantum groups was formulated in [(ref:Fim10)]. | ||
| - | Let $G$ be a compact quantum group, $X\subset\Irr(G)$, and $\pi\colon C(G)\to B(H)$ a $∗$-representation on a Hilbert space $H$. For $x\in\Irr(G)$, denote $u^x$ its representative acting on $H_x$ and put $U^x:= (\id\otimes\pi)u^x\in B(H_x)\otimes B(H)$. | + | Let $G$ be a compact quantum group, $X\subset\Irr(G)$, and $\pi\colon C(G)\to B(H)$ a $*$-representation on a Hilbert space $H$. For $x\in\Irr(G)$, denote $u^x$ its representative acting on $H_x$ and put $U^x:=(\id\otimes\pi)u^x\in B(H_x)\otimes B(H)$. |
| - | For $\epsilon>0$ we say that the unit vector $v\in H$ is $(X,\epsilon)$-invariant if for all $x\in X$ and all | + | For $\epsilon>0$ we say that the unit vector $v\in H$ is $(X,\epsilon)$**-invariant** if for all $x\in X$ and all non-zero $\eta\in H_x$ we have |
| - | non-zero $\eta\in H_x$ we have | + | |
| $$\|U^x(\eta\otimes v) − (\eta\otimes v)\| < \epsilon\|\eta\|.$$ | $$\|U^x(\eta\otimes v) − (\eta\otimes v)\| < \epsilon\|\eta\|.$$ | ||
| - | We say the representation $\pi$ almost contains invariant vectors if there are $(X,\epsilon)$-invariant | + | We say the representation $\pi$ **contains almost invariant vectors** if there are $(X,\epsilon)$-invariant |
| vectors for all finite subsets $X\subset\Irr(G)$ and all $\epsilon>0$. | vectors for all finite subsets $X\subset\Irr(G)$ and all $\epsilon>0$. | ||
| - | We say that $\Gamma=\hat G$ has **property (T)** if every representation $\pi$ almost containing an invariant vector contains an invariant vector, that is, there is $v\in H$ such that | + | We say that $\Gamma=\hat G$ has **property (T)** if every representation $\pi$ containing almost invariant vector contains an invariant vector, that is, there is $v\in H$ such that |
| $$U^x(\eta\otimes v) = \eta\otimes v$$ | $$U^x(\eta\otimes v) = \eta\otimes v$$ | ||
| for all $x\in\Irr(G)$ and all $\eta\in H_x$. | for all $x\in\Irr(G)$ and all $\eta\in H_x$. | ||
| + | |||
| + | ===== Results ===== | ||
| + | |||
| + | ==== Stability results ==== | ||
| + | |||
| + | |||
| + | * Suppose $H\subset G$. If $\hat G$ has (T), then also $\hat H$ has (T). [(ref:Fim10)] | ||
| + | |||
| + | ==== Examples ==== | ||
| + | |||
| + | The following quantum groups have the property (T). | ||
| + | |||
| + | The following quantum groups do not have the property (T). | ||
| + | |||
| + | * $\hat O_N^+$ for $N\ge 2$ (since $\Z_2^{*N}$ does not have (T)) [(ref:Fim10)] | ||
| + | * $\hat U_N^+$ for $N\ge 2$ (since the free group $\mathbb{F}_N$ does not have (T)) [(ref:Fim10)] | ||
| + | * $\hat S_N^+$ for $N\ge 2$ [(ref:Fim10)] | ||
| + | * Any free group | ||
| + | * Any infinite Abelian discrete quantum group (since it is [[amenability|amenable]]) | ||
| + | |||
| + | |||
| + | |||
| + | ===== Relation with other properties ===== | ||
| + | |||
| + | If $\Gamma=\hat G$ has (T), then | ||
| + | |||
| + | * $\Gamma$ is finitely generated [(ref:Fim10)] | ||
| + | * $\Gamma$ is [[Unimodularity|unimodular]] [(ref:Fim10)] | ||
| + | |||
| + | Discrete quantum group $\Gamma=\hat G$ has (T) if... | ||
| + | |||
| + | Discrete quantum group $\Gamma$ has the [[haagerup_property|Haagerup property]] and property (T) if and only if $\Gamma$ is [[compact_quantum_group#Finite quantum groups|finite]] [(ref:DFSW13)]. | ||
| ===== References ===== | ===== References ===== | ||
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| )] | )] | ||
| - | [(ref:BCF18>> | ||
| - | title : Topological generation and matrix models for quantum reflection groups | ||
| - | year : 2018 | ||
| - | author : Michael Brannan, Alexandru Chirvasitu, Amaury Freslon | ||
| - | url : https://arxiv.org/abs/1808.08611 | ||
| - | )] | ||
| - | [(ref:Chi15>> | + | [(ref:DFSW13>> |
| - | title : Residually finite quantum group algebras | + | author : Matthew Daws, Pierre Fima, Adam Skalski, Stuart White |
| - | journal : Journal of Functional Analysis | + | title : The Haagerup property for locally compact quantum groups |
| - | volume : 268 | + | journal : Journal für die reine und angewandte Mathematik |
| - | number : 11 | + | volume : 2016 |
| - | pages : 3508–3533 | + | number : 711 |
| - | year : 2015 | + | pages : 189–229 |
| - | doi : https://doi.org/10.1016/j.jfa.2015.01.013 | + | year : 2013 |
| - | url : http://www.sciencedirect.com/science/article/pii/S0022123615000373 | + | doi : 10.1515/crelle-2013-0113 |
| - | author : Alexandru Chirvasitu | + | url : https://doi.org/10.1515/crelle-2013-0113 |
| )] | )] | ||
| Line 59: | Line 86: | ||
| )] | )] | ||
| - | [(ref:Sol05>> | ||
| - | author : Piotr M. Sołtan | ||
| - | doi : 10.1215/ijm/1258138137 | ||
| - | journal : Illinois Journal of Mathematics | ||
| - | number : 4 | ||
| - | pages : 1245–1270 | ||
| - | publisher : Duke University Press | ||
| - | title : Quantum Bohr compactification | ||
| - | url : https://doi.org/10.1215/ijm/1258138137 | ||
| - | volume : 49 | ||
| - | year : 2005 | ||
| - | )] | ||
| ~~REFNOTES ref ~~ | ~~REFNOTES ref ~~ | ||
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