On permanence of regularity properties

We study a pair of [Formula: see text]-algebras [Formula: see text] by associating a ∗-homomorphism from [Formula: see text] to [Formula: see text] which allows an approximate left inverse to the ultrapower [Formula: see text]-algebra of [Formula: see text] as a completely positive map of order zero and show that important regularity properties related to the Elliott program pass from [Formula: see text] to [Formula: see text] in our setting.