no. 5
Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra
[Croissance polynomiale de groupes quantiques discrets, dimension topologique du dual et * -régularité de l’algèbre de Fourier]
D’Andrea, Alessandro1 ; Pinzari, Claudia1 ; Rossi, Stefano2
1 Dipartimento di Matematica Università degli Studi di Roma “La Sapienza” I-00185 Roma (Italy)
2 Dipartimento di Matematica Università degli Studi di Roma “Tor Vergata” I-00133 Roma (Italy)
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(G), c’est-à-dire que tout idéal fermé de C(G) a intersection dense avec A(G). En particulier, A(G) 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(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C * –norm.

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DOI : 10.5802/aif.3127
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

D’Andrea, Alessandro 1 ; Pinzari, Claudia 1 ; Rossi, Stefano 2

1 Dipartimento di Matematica Università degli Studi di Roma “La Sapienza” I-00185 Roma (Italy)
2 Dipartimento di Matematica Università degli Studi di Roma “Tor Vergata” I-00133 Roma (Italy)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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D’Andrea, Alessandro; Pinzari, Claudia; Rossi, Stefano. Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra. Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2003-2027. doi : 10.5802/aif.3127. https://aif.centre-mersenne.org/articles/10.5802/aif.3127/

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