Automatic equality of quasitraces and traces on C*-algebras

Determine whether for every C*-algebra A the equality QT(A) = T(A) holds, i.e., whether every normalised quasitrace on A is a trace. This is known to be true for exact C*-algebras by Haagerup’s theorem, but its validity in full generality remains unresolved.

Background

The paper studies regularity properties of ample diagonal pairs (D ⊂ A) and introduces diagonal comparison, relating it to dynamical comparison and strict comparison. Many of the main results assume QT(A) = T(A), meaning that all quasitraces on A are traces, to match trace-based dimension function comparisons.

The authors note that while QT(A) = T(A) is established for exact C*-algebras by Haagerup, it is a longstanding question whether this equality holds automatically for arbitrary C*-algebras. Resolving this would broaden the applicability of their equivalences without the need to assume exactness or impose QT(A)=T(A) explicitly.

References

It is a long-standing open problem whether this is actually automatic; it is indeed true when A is exact, due to the celebrated result of Haagerup [12].

— Diagonal comparison of ample C*-diagonals  (2410.05967 - Kopsacheilis et al., 2024) in Introduction, preceding Theorem B (Page 4)

It is an open question whether every 2-quasitrace on a $C*$-algebra is a trace (asked by Kaplansky).

— Maximal Algebraic Ideals in Nonunital $C^*$-Algebras  (2609.18840 - Liu et al., 16 Sep 2026) in Remark 2.3, Section 2, Preliminaries