====== Amenability ====== The property of amenability was originally introduced by John von Neumann for locally compact groups (see [[wp>Amenable group]]). This property was later generalized to the case of locally compact quantum groups [(ref:BT03)]. Sometimes amenabile C*-algebra is defined as a synonym for [[wp>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 $\chi=\sum_{i=1}^N u_{ii}$. [(ref: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 [(ref: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 $\Gamma/\Lambda$. [(ref:Cra17)] ==== Examples ==== * $\hat O_F^+$ for $F\in\GL(2,\C)$ (in particular $\widehat{\mathrm{SU}}_q(2)$) [(ref:Ban97)] * Dual of quantum automorphism group of four-dimensional C*-algebras (in particular $\hat S_4^+$) [(ref:Ban99b)] * $\hat O_N^*$ [(ref: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 $N>2$ [(ref:Ban97)] * $\hat U_F^+$ for $F$ arbitrary [(ref:Ban97)] * Dual of quantum automorphism group of $N$-dimensional C*-algebras for $N>4$ (in particular $\hat S_N^+$) [(ref: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 [[wp>Nuclear C*-algebra|nuclear]] [(ref:BT03)] * $L^\infty(G)$ is [[wp>Von_Neumann_algebra#Amenable_von_Neumann_algebras|injective]] [(ref:BT03)] * $\Gamma$ has the [[Haagerup property]] [(ref:DFSW13)] A discrete quantum group $\Gamma=\hat G$ is amenable if ===== Further reading ===== * Michael Brannan. //[[https://arxiv.org/abs/1605.01770|Approximation properties for locally compact quantum groups]]//, 2016. ===== References ===== [(ref:BT03>> author : Erik Bédos and Lars Tuset title : Amenability and Co-Amenability for Locally Compact Quantum Groups journal : International Journal of Mathematics volume : 14 number : 08 pages : 865–884 year : 2003 doi : 10.1142/S0129167X03002046 url : https://doi.org/10.1142/S0129167X03002046 )] [(ref:Ban97>> author : Teodor Banica title : Le Groupe Quantique Compact Libre U(n) journal : Communications in Mathematical Physics year : 1997 volume : 190 number : 1 pages : 143--172 url : http://dx.doi.org/10.1007/s002200050237 )] [(ref:Ban99a>> author : Teodor Banica title : Representations of compact quantum groups and subfactors journal : Journal für die reine und angewandte Mathematik year : 1999 volume : 509 pages : 167--198 url : https://doi.org/10.1515/crll.1999.509.167 )] [(ref:Ban99b>> author : Teodor Banica title : Symmetries of a generic coaction journal : Mathematische Annalen year : 1999 volume : 314 number : 4 pages : 763–780 doi : 10.1007/s002080050315 url : https://doi.org/10.1007/s002080050315 )] [(ref:BV10>> author : Teodor Banica and Roland Vergnioux title : Invariants of the half-liberated orthogonal group journal : Annales de l'Institut Fourier volume : 60 number : 6 year : 2010 pages : 2137–2164 doi : 10.5802/aif.2579 url : http://www.numdam.org/item/AIF_2010__60_6_2137_0 )] [(ref:Cra17>> author : Jason Crann title : On hereditary properties of quantum group amenability journal : Proceedings of the American Mathematical Society volume : 145 year : 2017 pages : 627–635 doi : 10.1090/proc/13365 url : https://doi.org/10.1090/proc/13365 )] [(ref:DFSW13>> author : Matthew Daws, Pierre Fima, Adam Skalski, Stuart White title : The Haagerup property for locally compact quantum groups journal : Journal für die reine und angewandte Mathematik volume : 2016 number : 711 pages : 189–229 year : 2013 doi : 10.1515/crelle-2013-0113 url : https://doi.org/10.1515/crelle-2013-0113 )] [(ref:Avi82>> url : http://www.jstor.org/stable/1998890 author : Daniel Avitzour journal : Transactions of the American Mathematical Society number : 2 pages : 423--435 title : Free Products of C*-Algebras volume : 271 year : 1982 } )] ~~REFNOTES ref ~~