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.