Tensorially Absorbing Inclusions of C*-algebras

When $\mathcal{D}$ is strongly self-absorbing we say an inclusion $B\subseteq A$ is $\mathcal{D}$-stable if it is isomorphic to the inclusion $B\otimes \mathcal{D}\subseteq A\otimes \mathcal{D}$. We give ultrapower characterizations and show that if a unital inclusion is $\mathcal{D}$-stable, then $\mathcal{D}$-stability can be exhibited for countably many intermediate C*-algebras concurrently. We show that such embeddings between $\mathcal{D}$-stable C*-algebras are point-norm dense in the set of all embeddings, and that every embedding between $\mathcal{D}$-stable C*-algebras is approximately unitarily equivalent to a $\mathcal{D}$-stable embedding. Examples are provided.