Abstract classification theorems for amenable $C^{\ast }$-algebras

Abstract

Operator algebras are subalgebras of the bounded operators on a Hilbert space. They divide into two classes: $C^{\ast }$-algebras and von Neumann algebras according to whether they are required to be closed in the norm or weak-operator topology, respectively. 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 ${\mathrm{II}}_1$ 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 $C^{\ast }$-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 $C^{\ast }$-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 ($\mathcal{Z}$-stability). These hypotheses are abstract, and there are deep tools for verifying the universal coefficient theorem and $\mathcal{Z}$-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 Carrión, 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 $C^{\ast }$-algebras en route to classification. The companion survey (White, in preparation) focuses on the role of the $\mathcal{Z}$-stability hypothesis, and the associated work on “regularity.”