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.
Categories: math.OA