Stable rank of the reduced 𝐶*-algebras of non-amenable Lie groups of type I

Sudo, Takahiro

doi:10.1090/S0002-9939-97-04034-3

Stable rank of the reduced $C^*$-algebras of non-amenable Lie groups of type I
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Proc. Amer. Math. Soc. 125 (1997), 3647-3654 Request permission

Abstract:

In this paper we show that stable rank of the reduced $C^{*}$-algebras of connected non-compact real semi-simple Lie groups is estimated by real rank of these groups. We extend this result to the case of connected reductive Lie groups and partially even to the case of connected non-amenable real Lie groups of type I. As a corollary, we show that the product formula of stable rank holds for locally compact, $\sigma$-compact non-amenable groups of type I.

References

Robert Boyer and Robert Martin, The regular group $C^*$-algebra for real-rank one groups, Proc. Amer. Math. Soc. 59 (1976), no. 2, 371–376. MR 476913, DOI 10.1090/S0002-9939-1976-0476913-8
J. Dixmier, $C^{*}$-Algebras, North-Holland, Amsterdam-New York-Oxford, 1962.
J. M. G. Fell, Weak containment and Kronecker products of group representations, Pacific J. Math. 13 (1963), 503–510. MR 155932
J. M. G. Fell and R. S. Doran, Representations of $^*$-algebras, locally compact groups, and Banach $^*$-algebraic bundles. Vol. 2, Pure and Applied Mathematics, vol. 126, Academic Press, Inc., Boston, MA, 1988. Banach $^*$-algebraic bundles, induced representations, and the generalized Mackey analysis. MR 936629, DOI 10.1016/S0079-8169(09)60018-0
Pierre de la Harpe and Alain Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 (1989), 158 (French, with English summary). With an appendix by M. Burger. MR 1023471
Eberhard Kaniuth, Group $C^\ast$-algebras of real rank zero or one, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1347–1354. MR 1164146, DOI 10.1090/S0002-9939-1993-1164146-1
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
Ronald L. Lipsman, The dual topology for the principal and discrete series on semisimple groups, Trans. Amer. Math. Soc. 152 (1970), 399–417. MR 269778, DOI 10.1090/S0002-9947-1970-0269778-X
Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
Takahiro Sudo and Hiroshi Takai, Stable rank of the $C^*$-algebras of nilpotent Lie groups, Internat. J. Math. 6 (1995), no. 3, 439–446. MR 1327158, DOI 10.1142/S0129167X95000158
—, Stable rank of the $C^{*}$-algebras of solvable Lie groups of type I, preprint (1996).