a nuclear c*-algebra is quasi-diagonal if there is a unital embedding into the ultraproduct of matrices. the concept is essential in the elliott classification program. the existence of such embeddings is shown by the tikusis-white-winter theorem (2015) for any stably finite, simple, separable, nuclear, unital c*-algebra satisfying the uct. thus an interesting question is to understand how unique are such embeddings given the existence.

we are able to answer this question by obtaining the corresponding kk-uniqueness theorem and applying the abstract classification approach. the major difficulty appears since the codomain of the map does not (separably) tensorially absorb the jiang-su algebra, and thus not covered by previous classification techniques. we will explain how to tackle the difficulty by looking at k-theoretical properties of a special c*-algebra, the paschke dual algebra, associated to the map.