A Characterization of Tracially Nuclear C*-algebras

TL;DR

This paper provides two characterizations of tracially nuclear C*-algebras, linking their structure to hyperfiniteness of their second dual and a weak* uniqueness property, with a focus on separable cases.

Contribution

It introduces two new characterizations of tracially nuclear C*-algebras, including a necessary and sufficient condition for separable algebras.

Findings

Finite summand of the second dual is hyperfinite

Characterization via a weak* uniqueness property

Necessary condition holds generally, sufficiency for separable algebras

Abstract

We give two characterizations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.