A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory

TL;DR

This paper explores the Atiyah-Segal completion theorem within KK-theory using a categorical approach, relating homological ideals to representation rings and extending results to groupoids and dynamical systems.

Contribution

It introduces a categorical perspective on the Atiyah-Segal theorem in KK-theory, connecting homological ideals with representation rings and extending to groupoid and dynamical system contexts.

Findings

Relates homological ideals to the augmentation ideal of the representation ring.

Extends the Atiyah-Segal completion theorem to groupoid KK-theory.

Studies the permanence of the Baum-Connes conjecture under group extensions.

Abstract

We investigate the homological ideal $\mathfrak{J}_G^H$, the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah-Segal completion theorem with the comparison of $\mathfrak{J}_G^H$ with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant $\mathrm{KK}$-theory, McClure's restriction map theorem, permanence property of the Baum-Connes conjecture under extensions of groups and a class of $\mathfrak{J}_G$-injective objects coming from $\mathrm{C}^*$-dynamical systems, continuous Rokhlin property.