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Kazhdan property (T)

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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