This paper shows that two natural ways of putting the Jiang–Su algebra Z into a particular “free” C*-algebra are not the same, even up to the usual notion of approximation used in operator algebras. In plain terms: the author finds a concrete target algebra where the two obvious embeddings of Z cannot be turned into one another by any sequence of unitary conjugations that converges in norm. This answers a question posed by Schafhauser, Tikuisis, and White.

The main counterexample uses the universal unital free product Z *_{C} Z, which is a kind of free combination of two copies of Z that keeps the unit element. The paper proves that the two canonical unital embeddings of Z into that free product are not approximately unitarily equivalent. The author also shows the same failure of uniqueness if Z is replaced by any unital MF algebra whose matrix sizes are prime powers. The free-product target is known to be quasidiagonal, which the paper cites as background (Boca 1997).

To explain the significance, approximate unitary equivalence means that one embedding can be moved to the other by conjugating with a sequence of unitary elements and that the conjugated images converge in operator norm. The negative result says no such sequence exists for the canonical pair in the free-product target. By contrast, the paper proves a positive uniqueness result under strong regularity assumptions on the target algebra: if the target is simple, unital, has a single trace (monotracial), and satisfies strict comparison of positive elements by that trace (properties often summarized by the term “selfless” in recent work), then any two unital embeddings of Z are strongly asymptotically unitarily equivalent. Strong asymptotic unitary equivalence means there is a norm-continuous path of unitaries starting at the identity so that, in the long run, conjugating by these unitaries makes the two embeddings agree on every element of Z.