Exotic group C * -algebras of simple Lie groups with real rank one
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2117-2136.

Exotic group C * -algebras are C * -algebras that lie between the universal and the reduced group C * -algebra of a locally compact group. We consider simple Lie groups G with real rank one and investigate their exotic group C * -algebras C L p+ * (G), which are defined through L p -integrability properties of matrix coefficients of unitary representations. First, we show that the subset of equivalence classes of irreducible unitary L p+ -representations forms a closed ideal of the unitary dual of these groups. This result holds more generally for groups with the Kunze–Stein property. Second, for every classical simple Lie group G with real rank one and every 2q<p, we determine whether the canonical quotient map C L p+ * (G)C L q+ * (G) has non-trivial kernel. Our results generalize, with different methods, recent results of Samei and Wiersma on exotic group C * -algebras of SO 0 (n,1) and SU(n,1). In particular, our approach also works for groups with property (T).

Les C * -algèbres exotiques des groupes sont des C * -algèbres, qui se situent entre la C * -algèbre universelle et la C * -algèbre réduite d’un groupe localement compact. Nous considérons des groupes de Lie simples G de rang réel un et nous étudions leurs C * -algèbres exotiques C L p+ * (G), qui sont définies par des propriétés d’intégrabilité L p des coefficients des représentations unitaires. Nous montrons que les classes d’équivalence de représentations L p+ unitaires irréductibles forment un idéal fermé du dual unitaire de ces groupes. Ce résultat vaut plus généralement pour les groupes avec la propriété de Kunze–Stein. Pour chaque groupe de Lie simple classique G de rang un et chaque 2q<p, nous déterminons si l’application canonique C L p+ * (G)C L q+ * (G) a un noyau non trivial. Nos résultats généralisent, avec des méthodes différentes, des résultats récents de Samei et Wiersma sur les C * -algèbres exotiques des groupes SO 0 (n,1) et SU(n,1). En particulier, notre approche s’applique également à des groupes avec la propriété (T).

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DOI: 10.5802/aif.3441
Classification: 22D25,  22E46,  43A90
Keywords: group C * -algebras, L p+ -representations, simple Lie groups
de Laat, Tim 1; Siebenand, Timo 1

1 Westfälische Wilhelms-Universität Münster Mathematisches Institut Einsteinstraße 62 48149 Münster (Germany)
License: CC-BY-ND 4.0
     author = {de Laat, Tim and Siebenand, Timo},
     title = {Exotic group $C^{*}$-algebras of simple {Lie} groups with real rank one},
     journal = {Annales de l'Institut Fourier},
     pages = {2117--2136},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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de Laat, Tim; Siebenand, Timo. Exotic group $C^{*}$-algebras of simple Lie groups with real rank one. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2117-2136. doi : 10.5802/aif.3441. https://aif.centre-mersenne.org/articles/10.5802/aif.3441/

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