Commutativity of central sequence algebras

TL;DR

This paper characterizes when the central sequence algebra of a separable C*-algebra is abelian, linking it to Fell's condition, and explores higher-dimensional analogues and properties of extensions.

Contribution

It provides a complete characterization of abelian central sequence algebras via Fell's condition and introduces a higher-dimensional analogue for subhomogeneity.

Findings

A separable C*-algebra has abelian central sequence algebra iff it satisfies Fell's condition.

Higher-dimensional analogue of Fell's condition characterizes subhomogeneity.

Non-trivial extensions by compact operators have non-abelian, non-type I central sequence algebras.

Abstract

The question of which separable C*-algebras have abelian central sequence algebras was raised and studied by Phillips ([Ph88]) and Ando-Kirchberg ([AK14]). In this paper we give a complete answer to their question: A separable C*-algebra $A$ has abelian central sequence algebra if and only if A satisfies Fell's condition. Moreover, we introduce a higher-dimensional analogue of Fell's condition and show that it completely characterizes subhomogeneity of central sequence algebras. In contrast, we show that any non-trivial extension by compact operators has not only non-abelian but not even residually type I central sequence algebra. In particular its central sequence algebra is not type I and not residually finite-dimensional (RFD). Our techniques extensively use properties of nilpotent elements in C*-algebras.