Journal of Lie Theory 27 (2017), No. 3, 831--844
Copyright Heldermann Verlag 2017
Harish-Chandra's Schwartz Algebras Associated with Discrete Subgroups of Semisimple Lie Groups
Faculty of Mathematics, The Weizmann Institute of Science, Herzl Street 234, Rehovot, Israel
We prove that the Harish-Chandra's Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced C*-algebra of the discrete subgroup. Then, we prove that the reduced C*-norm is controlled by the norm of the Harish-Chandra's Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.
Keywords: Harish-Chandra's Schwartz spaces, semisimple Lie groups, Harish-Chandra functions, Furstenberg boundary, property RD, K-theory, Baum-Connes conjecture.
MSC: 46H15, 43A90, 22E40; 22D20, 22D25, 46L80
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