A note on non-unital absorbing extensions

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.