Generalized quasidiagonality for extensions

Abstract

We generalize the notion of quasidiagonality, for extensions, allowing for the case where the canonical ideal has few projections. We prove that the pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal. We also provide a $K$-theory sufficient condition for generalized quasidiagonality of certain extensions of simple continuous-scale $C^{\ast }$-algebras, including certain continuous-scale hereditary $C^{\ast }$-subalgebras of the stabilized Jiang–Su algebra.