Countable degree-1 saturation of certain $C$*-algebras which are coronas of Banach algebras

Abstract

We study commutants modulo some normed ideal of $n$-tuples of operators which satisfy a certain approximate unit condition relative to the ideal. We obtain results about the quotient of these Banach algebras by their ideal of compact operators being $C$*-algebras which have the countable degree-1 saturation property in the model-theory sense of I. Farah and B. Hart. We also obtain results about quasicentral approximate units, multipliers and duality.