Embeddings into the ultrapower of the Jiang-Su algebra

Abstract

We study existence of embeddings into ultrapowers of the Jiang-Su algebra $\mathcal{Z}$ and the Razak-Jacelon algebra $\mathcal{W}$. More specifically, we show that the cone over any separable $C^*$-algebra embeds into the ultrapowers of both $\mathcal{Z}$ and $\mathcal{W}$. We also show that the result for $\mathcal{Z}$ generalizes to separable and exact continuous fields of $C^*$-algebras for which one of the fibers embeds into the ultrapower of $\mathcal{Z}$, if this fiber is suitably well-behaved.