The paper highlights a central question in the theory of traces on C*-algebras: whether amenability of a trace implies quasidiagonality. This problem is longstanding and remains unresolved in full generality.
Significant partial progress is known: by results of Tikuisis–White–Winter (extended by Schafhauser and Gabe), every faithful amenable trace on an exact C*-algebra satisfying the UCT is quasidiagonal. Brown–Carrion–White proved that any amenable trace on a cone C*-algebra is quasidiagonal, reflecting a homotopy flavor since cones are contractible.
This paper contributes homotopy-invariance results that imply additional positive cases. For instance, if either A or B is exact, A is homotopy dominated by B, and all amenable traces on B are quasidiagonal, then all amenable traces on A are quasidiagonal (Theorem 4.9 and Corollary 4.12). Despite these advances, the general question across all C*-algebras remains open.