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--- amenability [2019/09/16 14:27]
d.gromada created
+++ amenability [2021/11/23 11:56] (current)
@@ Line 6: / Line 6: @@
 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 ====
-  * $O_F^+$ 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 ====
   * $\hat O_F^+$ 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 =====
@@ Line 47: / Line 80: @@
 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>>
@@ Line 57: / Line 113: @@
 pages     : 2137–2164
 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 ~~

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