OP-01067 Solution claimed

Is every quasitrace on a C*-algebra a trace?

Quasitraces are meant in the sense of Blackadar and Handelman (2-quasitraces). Haagerup proved that every quasitrace on an exact C$^*$-algebra is a trace.

Problem. Is every quasitrace on a C$^*$-algebra a trace?

Source: B. Blackadar and D. Handelman (1982), and equivalently (Haagerup) I. Kaplansky's question (1951) whether every AW*-factor of type II₁ is a von Neumann algebra; listed as C*.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Keywords: AW*-algebras, C*-algebras, Peterson's operator algebra problems, quasitraces

References

@article{blackadar1982,
  author  = {Blackadar, Bruce and Handelman, David},
  title   = {Dimension functions and traces on {C}$^*$-algebras},
  journal = {J. Funct. Anal.},
  volume  = {45},
  number  = {3},
  pages   = {297--340},
  year    = {1982},
  doi     = {10.1016/0022-1236(82)90009-X},
}
@article{kaplansky1951b,
  author  = {Kaplansky, Irving},
  title   = {Projections in {B}anach algebras},
  journal = {Ann. of Math.},
  volume  = {53},
  number  = {2},
  pages   = {235--249},
  year    = {1951},
  doi     = {10.2307/1969540},
}
@misc{peterson-cs1,
  author       = {Peterson, Jesse},
  title        = {Open problems in operator algebras},
  howpublished = {Web page},
  year         = {2026},
  url          = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
  note         = {Problem C*.1; version of 15 May 2026},
}
@article{haagerup2014,
  author        = {Haagerup, Uffe},
  title         = {Quasitraces on exact {C}$^*$-algebras are traces},
  journal       = {C. R. Math. Acad. Sci. Soc. R. Can.},
  volume        = {36},
  number        = {2-3},
  pages         = {67--92},
  year          = {2014},
  eprint        = {1403.7653},
  archivePrefix = {arXiv},
}
@misc{schafhauser2025-cs1,
  author        = {Schafhauser, Christopher and Tikuisis, Aaron and White, Stuart},
  title         = {Nuclear {C}$^*$-algebras: 99 problems},
  year          = {2025},
  eprint        = {2506.10902},
  archivePrefix = {arXiv},
  note          = {Problem I; v2, 8 May 2026},
}

Resolution claims

A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness v1 GitHub ↗

OpenAI

We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C*-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C*-algebras can be properly infinite, even when one factor is the reduced free-group algebra C*_r(F_2).

Counterexample · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: math @ 7ef5c0c, build/paper.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01067, "Is every quasitrace on a C*-algebra a trace?", clopen.solutions.