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--- amenability [2019/10/25 06:41]
d.gromada
+++ amenability [2021/11/23 11:56] (current)
@@ Line 24: / Line 24: @@
 ==== Stability results ====
-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)]
+  * 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 ====
@@ Line 137: / Line 139: @@
 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 ~~

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