The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras

TL;DR

This paper advances the understanding of the Mackey bijection for complex reductive groups by embedding the associated C*-algebra into the reduced C*-algebra of the group, revealing new structural insights.

Contribution

It constructs an embedding of the motion group's C*-algebra into the reduced C*-algebra of G and characterizes the continuous field linked to the Mackey bijection.

Findings

Constructed an embedding of the motion group's C*-algebra into G's reduced C*-algebra.

Characterized the continuous field of reduced group C*-algebras associated with the Mackey bijection.

Provided a new characterization of the Mackey bijection using the embedding.

Abstract

The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive group G, between the tempered dual of G and the unitary dual of the associated Cartan motion group. We shall construct an embedding of the C*-algebra of the motion group into the reduced C*-algebra of G, and use it to characterize the continuous field of reduced group C*-algebras that is associated to the Mackey bijection. We shall also obtain a new characterization of the Mackey bijection using the same embedding.