====== 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 =====

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 $U^x:=(\id\otimes\pi)u^x\in B(H_x)\otimes B(H)$.

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
$$\|U^x(\eta\otimes v) − (\eta\otimes v)\| < \epsilon\|\eta\|.$$

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 $\epsilon>0$.

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
$$U^x(\eta\otimes v) = \eta\otimes v$$
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 =====


[(ref:BBCW17>>
title     : Kirchberg factorization and residual finiteness for discrete quantum groups
year      : 2017
author    : Angshuman Bhattacharya, Michael Brannan, Alexandru Chirvasitu, Shuzhou Wang
url       : https://arxiv.org/abs/1712.08682
)]


[(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:Fim10>>
author    : Pierre Fima
title     : Kazhdan's property T for discrete quantum groups
journal   : International Journal of Mathematics
volume    : 21
number    : 01
pages     : 47--65
year      : 2010
doi       : 10.1142/S0129167X1000591X
url       : https://doi.org/10.1142/S0129167X1000591X
)]


~~REFNOTES ref ~~