The Unitary Conjugation Groupoid as a Universal Mediator of the Baum--Connes Assembly Map

TL;DR

This paper introduces the unitary conjugation groupoid as a universal mediator that canonically factors the Baum--Connes assembly map through Morita equivalence, linking index theory and K-theory in groupoid models.

Contribution

It establishes the unitary conjugation groupoid as a universal intermediary for the Baum--Connes assembly map, extending index theory beyond Type I cases and relating it to various groupoid models.

Findings

Factorization of the Baum--Connes map through the unitary conjugation groupoid

Extension of $K_1$ index framework to non-Type I examples

Recovery of classical index pairing for the irrational rotation algebra

Abstract

We show that the Baum--Connes assembly map factors canonically through the unitary conjugation groupoid, which serves as a universal mediator among groupoid models that are Morita equivalent to a given transformation groupoid. This establishes a structural link between groupoid-based index theory and the Baum--Connes program at the level of K-theory. Building on our previous development of unitary conjugation groupoids and their associated index theory, we extend the $K_1$ index framework beyond the Type I setting to non-Type I examples, including the irrational rotation algebra and amenable crossed products. Using Morita equivalence, we relate unitary conjugation groupoids to transformation and action groupoids, enabling the transfer of descent-type index constructions to these settings. Our main result shows that, among all groupoid realizations that are Morita equivalent to a transformation groupoid, the factorization through the unitary conjugation groupoid is canonical at the level of K-theory. This identifies the unitary conjugation groupoid as a universal intermediary for the Baum--Connes assembly map. As applications, we recover the classical index pairing with the tracial state for the irrational rotation algebra in the sense of Connes, and we prove that for amenable crossed products the descent construction agrees with the analytic Baum--Connes assembly map under Morita equivalence. These results provide a conceptual interpretation of the assembly map in terms of internal symmetries of crossed product algebras and suggest a unified framework connecting Fredholm-type index data with equivariant K-theory via groupoid methods.