Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

D’Andrea, Alessandro; Pinzari, Claudia; Rossi, Stefano

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2003-2027.

Banica et Vergnioux ont montré que le groupe quantique discret dual d’un groupe de Lie compact et simplement connexe a croissance polynomiale de degré égal à la dimension réelle de la variété. On étend ce résultat aux groupes compactes quelconques et à leur dimension topologique, en la reliant à la dimension de Gelfand–Kirillov d’une algèbre. De plus, on prouve que la croissance polynomiale, pour un groupe quantique compact de Kac $G$, implique la ${}^{*}$-régularité de l’algèbre de Fourier $A\left(G\right)$, c’est-à-dire que tout idéal fermé de $C\left(G\right)$ a intersection dense avec $A\left(G\right)$. En particulier, $A\left(G\right)$ admet une unique norme $C^{*}$.

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand–Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group $G$ of Kac type implies ${}^{*}$–regularity of the Fourier algebra $A\left(G\right)$, that is every closed ideal of $C\left(G\right)$ has a dense intersection with $A\left(G\right)$. In particular, $A\left(G\right)$ has a unique $C^{*}$–norm.

Classification : 58B32, 46L65, 43A20, 16P90
Keywords: quantum group, topological dimension, polynomial growth, Fourier algebra
Mot clés : Groupe quantique, dimension topologique, croissance polynomiale, algèbre de Fourier

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