amenability
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| - The cannonical map $C_{\rm u}(G)\to C_{\rm r}(G)$ is an isomorphism | - 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**.š | + | 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 ===== | ===== 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 ==== | ==== Examples ==== | ||
| - | * $O_F^+$ for $F\in\GL(2,\C)$ (in particular $\widehat{\mathrm{SU}}_q(2)$) [(ref:Ban97)] | + | * $\hat O_F^+$ for $F\in\GL(2,\C)$ (in particular $\widehat{\mathrm{SU}}_q(2)$) [(ref:Ban97)] |
| - | * $O_N^*$ [(ref:BV10)] | + | * 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 ==== | ==== Non-examples ==== | ||
| * $\hat O_F^+$ for $F\in\GL(N,\C)$ (in particular $\hat O_N^+$) for $N>2$ [(ref:Ban97)] | * $\hat O_F^+$ 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 ===== | ===== Relation with other properties ===== | ||
| Line 29: | Line 47: | ||
| If $\Gamma=\hat G$ is an amenable discrete quantum group then | 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)] | * $C_{\rm u}(G)$ 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)] | + | * $L^\infty(G)$ 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 | 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 ===== | ===== References ===== | ||
| Line 58: | Line 80: | ||
| url : http://dx.doi.org/10.1007/s002200050237 | 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>> | [(ref:BV10>> | ||
| Line 68: | Line 113: | ||
| pages : 2137–2164 | pages : 2137–2164 | ||
| doi : 10.5802/aif.2579 | doi : 10.5802/aif.2579 | ||
| - | url : http://www.numdam.org/item/AIF_2010__60_6_2137_0} | + | 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 ~~ | ~~REFNOTES ref ~~ | ||
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