amenability
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| - | ====== 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 | ||
| - | - ... | ||
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| - | If one of those equivalent conditions is satisfied, we call $\Gamma$ **amenable** and $G$ **co-amenable**. | ||
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| - | ===== Results ===== | ||
| - | |||
| - | ==== Examples ==== | ||
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| - | * $O_F^+$ for $F\in\GL(2,\C)$ (in particular $\widehat{\mathrm{SU}}_q(2)$) [(ref:Ban97)] | ||
| - | * $O_N^*$ [(ref:BV10)] | ||
| - | |||
| - | ==== Non-examples ==== | ||
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| - | * $\hat O_F^+$ for $F\in\GL(N,\C)$ (in particular $\hat O_N^+$) for $N>2$ [(ref:Ban97)] | ||
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| - | ===== References ===== | ||
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| - | [(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: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} | ||
| - | )] | ||
| - | |||
| - | ~~REFNOTES ref ~~ | ||
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