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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

Relation with other properties

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

$\Gamma$ is finitely generated [Fim10]
$\Gamma$ is of Kac type [Fim10]

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

References

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

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Kazhdan property (T)

Definition

Results

Relation with other properties

References

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