Kaplansky's quasitrace problem
Kaplansky's quasitrace problem: is every normalized 2-quasitrace on a unital C*-algebra a tracial state? The answer is no: a separable unital C*-algebra has quasitraces, and every one of them fails additivity by at least 1/144 on one fixed pair of positive elements. The proof: a defect lemma from two relations between finitely many elements, bundles over products of projective spaces with a Chern class obstruction, generic expanding maps normalized by a determinant potential, and an algebra of weighted shifts. Proved in full from two named hypotheses: the existence theorem for quasitraces of Cuntz, Blackadar and Handelman, and standard facts on smooth manifolds. The course is an exposition of OpenAI's proof (2026), with a sharper constant.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Quasitraces and a uniform additivity defect
- Expanding maps and a continuous normalization
- Bundle tests with a Chern class obstruction
- Weighted shifts and the construction
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- The Gysin sequence and projective splitting · Characteristic classes
- Chern classes and the integral universal ring · Characteristic classes
- Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
- C*-algebras and positivity (course Foundations of von Neumann algebras); Chern classes and the cohomology of projective bundles (course Characteristic classes).