A Mackey embedding for reduced C*-algebras of real reductive groups

Abstract

The purpose of this paper is construct an embedding of the C*-algebra of the Cartan motion group of a real reductive group G into the reduced C*-algebra of G itself. The embedding has a number of applications: we shall use it to characterize the Mackey bijection from the tempered dual of G into the unitary dual of the motion group; to characterize the continuous field of reduced group C*-algebras arising from the contraction of G to its Cartan motion group; and to characterize the Connes-Kasparov assembly map in operator K-theory. Our results continue and complete a project that was begun several years ago by the last two authors, who considered the case of complex groups. In the real case, detailed information from the theory of R-groups is used in the construction.

Theorems & Definitions (79)

• Lemma 2.2.2: See Higson08
• Definition 2.3.1
• Lemma 2.3.2
• Definition 3.4.2
• Lemma 3.4.3
• Lemma 3.5.2: Induction in stages; see e.g. VoganGreenBook
• Lemma 3.5.3
• Theorem 3.6.5
• Remark 3.6.6
• proof : Proof of Theorem \ref{['thm-intertwiners-constant-in-some-directions']}