Amenability

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 $m(\omega\otimes\id)\Delta=\omega(1)m$ .
$G$ is quantum injective. That is, there exists a conditional expectation $E\colon B(L^2(G))\to l^\infty(\Gamma)$ with $E(L^\infty(G))=\C$ (equivalently $E(L^\infty(G))\subset Z(l^\infty(\Gamma))$ )
The counit $\epsilon$ of $\Pol G$ extends to $C_{\rm r}(G)$ .
The Haar state $h$ is faithful on $C_{\rm u}(G)$
The cannonical map $C_{\rm u}(G)\to C_{\rm r}(G)$ is an isomorphism

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

Some special cases

Suppose that $\Gamma$ is unimodular. Then $\Gamma$ is amenable if and only if $L^\infty(G)$ is injective.

Suppose that $G$ is a compact matrix quantum group with fundamental representation $u\in M_N(C(G))$ . Then $\Gamma=\hat G$ is amenable if and only if $N$ is in the spectrum of $\mathop{\rm Re}\chi\in C(G)$ , where $\chi=\sum_{i=1}^N u_{ii}$ . [Ban99a]

Results

Stability results

If $H_1$ and $H_2$ are co-amenable, then $H_1\times H_2$ is co-amenable. (Proof: Denote by $h_1$ and $h_2$ the Haar states and by $\epsilon_1$ and $\epsilon_2$ the counits defined on $C(H_1)$ resp. $C(H_2)$ . Take $A:=C(H_1)\otimes_{\rm min}C(H_2)$ . We check that $\epsilon:=\epsilon_1\otimes\epsilon_2$ is a counit of $H_1\times H_2$ and $h:=h_1\otimes h_2$ is the Haar state of $H_1\times H_2$ . From [Avi82] (Appendix) it follows that $h$ is faithful on $A$ .)
If a discrete quantum group $\Gamma$ is amenable than every its quantum subgroup $\Lambda$ is amenable (we have $C^*(\Lambda)\subset C^*(\Gamma)$ , hence if $h$ is faithful on $C^*(\Gamma)$ then it is faithful also on $C^*(\Lambda)$ ). Moreover, $\Gamma$ is amenable if and only if $\Lambda$ is amenable and $\Gamma$ acts amenably on the homogeneous space $\Gamma/\Lambda$ . [Cra17]

Examples

$\hat O_F^+$ for $F\in\GL(2,\C)$ (in particular $\widehat{\mathrm{SU}}_q(2)$ ) [Ban97]
Dual of quantum automorphism group of four-dimensional C*-algebras (in particular $\hat S_4^+$ ) [Ban99b]
$\hat O_N^*$ [BV10]
Any finite quantum group
Any Abelian quantum group (i.e. a dual of a compact group)

Non-examples

$\hat O_F^+$ for $F\in\GL(N,\C)$ (in particular $\hat O_N^+$ ) for $N>2$ [Ban97]
$\hat U_F^+$ for $F$ arbitrary [Ban97]
Dual of quantum automorphism group of $N$ -dimensional C*-algebras for $N>4$ (in particular $\hat S_N^+$ ) [Ban99b]
Any countable discrete group containing a free subgroup on two generators

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]
$\Gamma$ has the Haagerup property [DFSW13]

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.

[Ban99a] Teodor Banica, 1999. Representations of compact quantum groups and subfactors. Journal für die reine und angewandte Mathematik, 509, pp.167–198.

[Avi82] Daniel Avitzour, 1982. Free Products of C*-Algebras. Transactions of the American Mathematical Society, 271(2), pp.423–435.

[Cra17] Jason Crann, 2017. On hereditary properties of quantum group amenability. Proceedings of the American Mathematical Society, 145, pp.627–635.

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

[Ban99b] Teodor Banica, 1999. Symmetries of a generic coaction. Mathematische Annalen, 314(4), pp.763–780.

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

[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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