A weak homotopy equivalence type result related to Kirchberg algebras

TL;DR

This paper establishes a weak homotopy equivalence between certain topological groups related to Kirchberg algebras, aiding in their classification and revealing isomorphisms between K-groups and KK-groups.

Contribution

It proves a weak homotopy equivalence between the unitary group of the asymptotic centralizer and the loop group of automorphisms for Kirchberg algebras, advancing classification methods.

Findings

Weak homotopy equivalence between topological groups for Kirchberg algebras

Isomorphism between K-groups of the asymptotic centralizer and KK-groups

Foundation for classifying poly-Z group actions on Kirchberg algebras

Abstract

We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$-groups of the continuous asymptotic centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra.