Extensions of quasidiagonal $C^*$-algebras and controlling the $K_0$-map of embeddings

TL;DR

This paper investigates the Blackadar-Kirchberg conjecture for extensions of specific classes of quasidiagonal, nuclear, separable $C^*$-algebras satisfying the UCT, focusing on the behavior of the $K_0$-map in embeddings.

Contribution

It establishes conditions under which the conjecture holds for extensions, particularly when the ideal belongs to a class closed under local approximations and includes various important subclasses.

Findings

The conjecture holds for extensions with ideals in the specified class.

The class includes all separable ASH algebras and certain simple, unital $C^*$-algebras.

The results apply to crossed products with the integers.

Abstract

We study the validity of the Blackadar-Kirchberg conjecture for extensions of separable, nuclear, quasidiagonal $C^*$-algebras that satisfy the UCT. More specifically, we show that the conjecture for the extension has an affirmative answer if the ideal lies in a class of $C^*$-algebras that is closed under local approximations and contains all separable ASH algebras, as well as certain classes of simple, unital $C^*$-algebras and crossed products of unital $C^*$-algebras with the integers.