Nonsplitting in Kirchberg's ideal-related KK-theory

TL;DR

This paper investigates the splitting properties of the universal coefficient theorem in Kirchberg's ideal-related KK-theory, showing it generally does not split and exploring implications for classifying purely infinite C*-algebras.

Contribution

It demonstrates that Bonkat's UCT does not split in general and introduces methods to analyze the K-theory complexity needed for classifying certain C*-algebras.

Findings

Bonkat's UCT does not split in general

Methods to analyze K-theory complexity for classification

Implications for purely infinite C*-algebras with one ideal

Abstract

A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal.