amenability
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amenability [2019/10/09 12:52] d.gromada [Relation with other properties] |
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| ==== Stability results ==== | ==== Stability results ==== | ||
| - | If a discrete quantum group $\Gamma$ is amenable than every its quantum subgroup is amenable. [(ref:Cra17)] | + | |
| + | * 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 ==== | ||
| Line 46: | Line 48: | ||
| * $C_{\rm u}(G)$ 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 | ||
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| url : https://doi.org/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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