Exotic group $C^{*}$-algebras of simple Lie groups with real rank one

de Laat, Tim; Siebenand, Timo

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2117-2136.

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 $C_{L^{p+}}^{*}\left(G\right)$, 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 $2\le q<p\le \infty$, nous déterminons si l’application canonique $C_{L^{p+}}^{*}\left(G\right)\twoheadrightarrow C_{L^{q+}}^{*}\left(G\right)$ 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 ${\mathrm{SO}}_0(n,1)$ et $\mathrm{SU}(n,1)$. En particulier, notre approche s’applique également à des groupes avec la propriété (T).

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 $C_{L^{p+}}^{*}\left(G\right)$, 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 $2\le q<p\le \infty$, we determine whether the canonical quotient map $C_{L^{p+}}^{*}\left(G\right)\twoheadrightarrow C_{L^{q+}}^{*}\left(G\right)$ has non-trivial kernel. Our results generalize, with different methods, recent results of Samei and Wiersma on exotic group $C^{*}$-algebras of ${\mathrm{SO}}_0(n,1)$ and $\mathrm{SU}(n,1)$. In particular, our approach also works for groups with property (T).

Classification : 22D25, 22E46, 43A90
Keywords: group $C^*$-algebras, $L^{p+}$-representations, simple Lie groups
Mot clés : $C^*$-algèbres des groupes, représentations $L^{p+}$, groupes de Lie simples

[1] Baum, Paul F.; Guentner, Erik P.; Willett, Rufus Expanders, exact crossed products, and the Baum–Connes conjecture, Ann. $K$-Theory, Volume 1 (2016) no. 2, pp. 155-208 | DOI | MR | Zbl

[2] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008 | DOI | MR | Zbl

[3] Brown, Nathanial P.; Guentner, Erik P. New ${\mathrm{C}}^{*}$-completions of discrete groups and related spaces, Bull. Lond. Math. Soc., Volume 45 (2013) no. 6, pp. 1181-1193 | DOI | MR | Zbl

[4] Buss, Alcides; Echterhoff, Siegfried; Willett, Rufus Exotic crossed products, Operator algebras and applications. The Abel symposium 2015 took place on the ship Finnmarken, part of the Coastal Express Line (the Norwegian Hurtigruten), from Bergen to the Lofoten Islands, Norway, August 7-11, 2015 (Abel Symposia), Volume 12, Springer, 2016, pp. 61-108 | MR | Zbl

[5] Buss, Alcides; Echterhoff, Siegfried; Willett, Rufus Exotic crossed products and the Baum–Connes conjecture, J. Reine Angew. Math., Volume 740 (2018), pp. 111-159 | DOI | MR | Zbl

[6] Cowling, Michael G. The Kunze–Stein phenomenon, Ann. Math., Volume 107 (1978) no. 2, pp. 209-234 | DOI | MR | Zbl

[7] Cowling, Michael G. Sur les coefficients des représentations unitaires des groupes de Lie simples, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976–1978), II (Lecture Notes in Mathematics), Volume 739, Springer, 1979, pp. 132-178 | MR | Zbl

[8] van Dijk, Gerrit Introduction to harmonic analysis and generalized Gelfand pairs, De Gruyter Studies in Mathematics, 36, Walter de Gruyter, 2009 | DOI | MR | Zbl

[9] Dixmier, Jacques $C^{*}$-algebras, North-Holland Mathematical Library, 15, North-Holland Publishing Company, 1977 (Translated from the French by Francis Jellett) | MR | Zbl

[10] Gangolli, Ramesh; Varadarajan, Veeravalli S. Harmonic analysis of spherical functions on real reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 101, Springer, 1988 | DOI | MR | Zbl

[11] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, 2001 (corrected reprint of the 1978 original) | DOI | MR | Zbl

[12] Howe, Roger On a notion of rank for unitary representations of the classical groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 223-331 | MR

[13] Kaliszewski, Steven P.; Landstad, Magnus B.; Quigg, John Exotic group $C^{*}$-algebras in noncommutative duality, New York J. Math., Volume 19 (2013), pp. 689-711 | MR | Zbl

[14] Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, 36, Princeton University Press, 1986 | DOI | MR | Zbl

[15] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser, 2002 | MR | Zbl

[16] Kostant, Bertram On the existence and irreducibility of certain series of representations, Bull. Am. Math. Soc., Volume 75 (1969), pp. 627-642 | DOI | MR | Zbl

[17] Kunze, Ray A.; Stein, Elias M. Uniformly bounded representations and harmonic analysis of the $2\times 2$ real unimodular group, Am. J. Math., Volume 82 (1960), pp. 1-62 | DOI | MR | Zbl

[18] Li, Jian-Shu The minimal decay of matrix coefficients for classical groups, Harmonic analysis in China (Cheng, Minde et al., eds.) (Mathematics and its Applications), Volume 327, Kluwer Academic Publishers, 1995, pp. 146-169 | MR | Zbl

[19] Li, Jian-Shu; Zhu, Chen-Bo On the decay of matrix coefficients for exceptional groups, Math. Ann., Volume 305 (1996) no. 2, pp. 249-270 | DOI | MR | Zbl

[20] Nebbia, Claudio Groups of isometries of a tree and the Kunze6-Stein phenomenon, Pac. J. Math., Volume 133 (1988) no. 1, pp. 141-149 | DOI | MR | Zbl

[21] Oh, Hee Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. Fr., Volume 126 (1998) no. 3, pp. 355-380 | DOI | Numdam | MR | Zbl

[22] Oh, Hee Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., Volume 113 (2002) no. 1, pp. 133-192 | DOI | MR | Zbl

[23] Okayasu, Rui Free group $C^{*}$-algebras associated with ${\ell }_p$, Int. J. Math., Volume 25 (2014) no. 7, 1450065 | DOI | MR | Zbl

[24] Samei, Ebrahim; Wiersma, Matthew Exotic $C^{*}$-algebras of geometric groups (2018) (https://arxiv.org/abs/1809.07007)

[25] Shalom, Yehuda Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier, Volume 50 (2000) no. 3, pp. 833-863 | DOI | Numdam | MR | Zbl

[26] Shalom, Yehuda Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. Math., Volume 152 (2000) no. 1, pp. 113-182 | DOI | MR | Zbl

[27] Veca, Alessandro The Kunze–Stein Phenomenon, Ph. D. Thesis, University of New South Wales, Sydney, Australia (2002)

[28] Wiersma, Matthew $L^p$-Fourier and Fourier–Stieltjes algebras for locally compact groups, J. Funct. Anal., Volume 269 (2015) no. 12, pp. 3928-3951 | DOI | MR | Zbl

[29] Wiersma, Matthew Constructions of exotic group $C^{*}$-algebras, Ill. J. Math., Volume 60 (2016) no. 3-4, pp. 655-667 | MR | Zbl

[30] Yokota, Ichiro Exceptional Lie groups (2009) (https://arxiv.org/abs/0902.0431)