Perturbations of nuclear C*-algebras

TL;DR

This paper proves that close separable nuclear C*-algebras are unitarily conjugate, confirming a conjecture by Kadison and Kastler, and explores related embeddings and classifications.

Contribution

It establishes the Kadison-Kastler conjecture for separable nuclear C*-algebras and develops methods for embeddings and classification of these algebras.

Findings

Close separable nuclear C*-algebras are unitarily conjugate

Embeddings from near inclusions are possible for certain nuclear algebras

Improved characterizations of algebra types relevant to classification

Abstract

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.