Traceless AF embeddings and unsuspended $E$-theory

TL;DR

This paper establishes that for traceless C*-algebras, quasidiagonality and AF embeddability are equivalent and characterizes these properties via the primitive ideal space, also linking to when E-theory can be unsuspended for nuclear C*-algebras.

Contribution

It provides a new characterization of quasidiagonality and AF embeddability for traceless C*-algebras and relates this to the unsuspension of E-theory in nuclear cases.

Findings

Quasidiagonality and AF embeddability are equivalent for traceless C*-algebras.

Characterization of these properties via primitive ideal space.

Conditions for unsuspending E-theory in nuclear C*-algebras.

Abstract

I show that quasidiagonality and AF embeddability are equivalent properties for traceless $\mathrm C^\ast$-algebras and are characterised in terms of the primitive ideal space. For nuclear $\mathrm C^\ast$-algebras the same characterisation determines when Connes and Higson's $E$-theory can be unsuspended.