A note on non-unital absorbing extensions

TL;DR

This paper revises a theorem on non-unital absorbing extensions of C*-algebras, providing a corrected formulation and exploring implications for classification theory.

Contribution

It corrects a flawed proof by Elliott and Kucerovsky, offering an equivalent condition for nuclear absorption with mild assumptions.

Findings

Counterexample to the original theorem

New equivalent formulation of nuclear absorption

Implications for classification results

Abstract

Elliott and Kucerovsky stated that a non-unital extension of separable $C^\ast$-algebras with a stable ideal, is nuclearly absorbing if and only if the extension is purely large. However, their proof was flawed. We give a counter example to their theorem as stated, but establish an equivalent formulation of nuclear absorption under a very mild additional assumption to being purely large. In particular, if the quotient algebra is non-unital, then we show that the original theorem applies. We also examine how this effects results in classification theory.