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Amenability

The property of amenability was originally introduced by John von Neumann for locally compact groups (see Amenable group). This property was later generalized to the case of locally compact quantum groups [BT03]. Sometimes amenabile C*-algebra is defined as a synonym for Nuclear C*-algebra.

Definition

Let $G$ be a compact quantum group and $\Gamma:=\hat G$ its discrete dual. The following are equivalent

There exists a left-invariant mean on $\Gamma$ . That is, there is a state $m\in l^\infty(\Gamma)^*$ such that for all $\omega\in l^1(\Gamma)$ we have
$m(\omega\otimes\id)\Delta=\omega(1)m.$

If one of those equivalent conditions is satisfied, we call $\Gamma$ amenable and $G$ co-amenable.

Results

Examples

$O_F^+$ for $F\in\GL(2,\C)$ (in particular $\widehat{\mathrm{SU}}_q(2)$ ) [Ban97]
$O_N^*$ [BV10]

Non-examples

$\hat O_F^+$ for $F\in\GL(N,\C)$ (in particular $\hat O_N^+$ ) for $N>2$ [Ban97]

Relation with other properties

If $\Gamma=\hat G$ is an amenable discrete quantum group then

$C_{\rm u}(G)$ and $C_{\rm r}(G)$ are nuclear [BT03]
$L^\infty(G)$ is injective [BT03]

A discrete quantum group $\Gamma=\hat G$ is amenable if

References

[BT03] Erik Bédos and Lars Tuset, 2003. Amenability and Co-Amenability for Locally Compact Quantum Groups. International Journal of Mathematics, 14(08), pp.865–884.

[Ban97] Teodor Banica, 1997. Le Groupe Quantique Compact Libre U(n). Communications in Mathematical Physics, 190(1), pp.143–172.

[BV10] Teodor Banica and Roland Vergnioux, 2010. Invariants of the half-liberated orthogonal group. Annales de l'Institut Fourier, 60(6), pp.2137–2164.

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