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 one of the $99$ problems in the list of Schafhauser, Tikuisis, and White. By contrast, we prove that any two such embeddings into a simple, unital, monotracial $\mathrm{C}^\ast$-algebra with strict comparison by the unique trace are strongly asymptotically unitarily equivalent.