Unital embeddings of the Jiang--Su algebra are not unique

Abstract

We show that the canonical embeddings of the Jiang--Su algebra $\mathcal{Z}$ into the universal unital free product $\mathcal{Z}{\ast }_{\mathbb{C}}\mathcal{Z}$ are not approximately unitarily equivalent. This answers a question appearing in one of the $99$ problems list of Schafhauser, Tikuisis, and White. By contrast, we prove that any two such embeddings into a selfless ${\mathrm{C}}^{\ast }$-algebra are strongly asymptotically unitarily equivalent.

arXiv commentv2: 21 pp, minor changes, added uniqueness of embeddings in the purely infinite case