Extensions by simple $C^*$-algebras -- Quasidiagonal extensions

TL;DR

This paper characterizes when essential extensions of certain $C^*$-algebras are approximately unitarily equivalent, computes quasidiagonality conditions, and explores the nature of trivial and non-trivial extensions.

Contribution

It provides a classification of essential extensions via $KL$-theory and determines quasidiagonality criteria for these extensions.

Findings

Extensions are classified by $KL(A, M(B)/B)$ when $A$ satisfies UCT.

Quasidiagonal extensions can be non-trivial.

Exact conditions for quasidiagonality of extensions are established.

Abstract

Let $A$ be an amenable separable \CA and $B$ be a non-unital but $\sigma$-unital simple \CA with continuous scale. We show that two essential extensions $\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [\tau_1]=[\tau_2] {\rm in} KL(A, M(B)/B). $$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B).$ Using $KL(A, M(B)/B),$ we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.