Kazhdan property (T)

This property was originally formulated by David Kazhdan for locally compact groups (see 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 [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). [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)) [Fim10]
$\hat U_N^+$ for $N\ge 2$ (since the free group $\mathbb{F}_N$ does not have (T)) [Fim10]
$\hat S_N^+$ for $N\ge 2$ [Fim10]
Any free group
Any infinite Abelian discrete quantum group (since it is amenable)

Relation with other properties

If $\Gamma=\hat G$ has (T), then

$\Gamma$ is finitely generated [Fim10]
$\Gamma$ is unimodular [Fim10]

Discrete quantum group $\Gamma=\hat G$ has (T) if…

Discrete quantum group $\Gamma$ has the Haagerup property and property (T) if and only if $\Gamma$ is finite [DFSW13].

References

[Fim10] Pierre Fima, 2010. Kazhdan's property T for discrete quantum groups. International Journal of Mathematics, 21(01), pp.47–65.

[DFSW13] Matthew Daws, Pierre Fima, Adam Skalski, Stuart White, 2013. The Haagerup property for locally compact quantum groups. Journal für die reine und angewandte Mathematik, 2016(711), pp.189–229.

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