Abstract classification theorems for amenable C*-algebras

TL;DR

This paper reviews the classification of amenable C*-algebras, highlighting recent abstract methods that connect to von Neumann algebra results without requiring internal approximation structures.

Contribution

It introduces a new abstract approach to the unital classification theorem for amenable C*-algebras, bypassing the need for internal approximation structures.

Findings

Unified framework connecting C*-algebra classification to von Neumann algebra results

Simplified verification of regularity hypotheses like Jiang-Su stability

Enhanced understanding of the history and context of the classification theorem

Abstract

In the 1970s Alain Connes identified the appropriate notion of amenabilty for von Neumann algebras, and used it to obtain a deep internal finite dimensional approximation structure for these algebras. This structure is exactly what is needed for classification, and one of many consequences of Connes' theorem is the uniqueness of amenable II$_1$ factors, and later a complete classification of all simple amenable von Neumann algebras acting on separable Hilbert spaces. The Elliott classification programme aims for comparable structure and classification results for $C^*$-algebras using operator $K$-theory and traces. The definitive unital classification theorem was obtained in 2015. This is a combination of the Kirchberg--Phillips theorem and the large scale activity in the stably finite case by numerous researchers over the previous 15--20 years. It classifies unital simple separable amenable $C^*$-algebras satisfying two extra hypotheses: a universal coefficient theorem which computes $KK$-theory in terms of $K$-theory and a regularity hypothesis excluding exotic high dimensional behaviour. Today the regularity hypothesis can be described in terms of tensor products (Jiang-Su stability). These hypotheses are abstract, and there are deep tools for verifying the universal coefficient theorem and Jiang-Su stability in examples. This article describes the unital classification theorem, its history and context, together with the new abstract approach to this result developed in collaboration with Carri\'on, Gabe, Schafhauser and Tikuisis. This method makes a direct connection to the von Neumann algebraic results, and does not need to obtain any kind of approximation structure inside $C^*$-algebras en route to classification. A companion survey will focus on the role of the Jiang-Su stability hypothesis, and the associated work on "regularity."