Harish-Chandra's Schwartz Algebras Associated with Discrete Subgroups of Semisimple Lie Groupsi

摘要

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.