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

TL;DR

This paper constructs an embedding of the C*-algebra of a real reductive group's Cartan motion group into the reduced C*-algebra of the group, enabling new characterizations in representation theory and operator K-theory.

Contribution

It introduces a novel embedding for real reductive groups, extending previous work on complex groups, with applications to the Mackey bijection and the Connes-Kasparov map.

Findings

Characterizes the Mackey bijection using the embedding.

Describes the continuous field of reduced C*-algebras from contraction.

Provides insights into the Connes-Kasparov assembly map.

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.