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amenability

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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

  1. 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 $m(\omega\otimes\id)\Delta=\omega(1)m$.
  2. $G$ 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 $E(L^\infty(G))\subset Z(l^\infty(\Gamma))$)
  3. The counit $\epsilon$ of $\Pol G$ extends to $C_{\rm r}(G)$.
  4. The Haar state $h$ is faithful on $C_{\rm u}(G)$
  5. 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 a discrete quantum group $\Gamma$ is amenable than every its quantum subgroup is amenable. [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

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

Further reading

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.
[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.
amenability.1570625525.txt.gz · Last modified: 2021/11/23 11:56 (external edit)

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