Every 2-quasitrace is a trace
Operator Algebras
2025-02-03 v2
Abstract
A heretofore longstanding open question of Kaplansky was, "Is every Type II_1 AW*-factor a von Neumann algebra?" In this paper, we answer this question in the affirmative. As a consequence, we establish that every 2-quasitrace on a unital C*-algebra is a trace.
Keywords
Cite
@article{arxiv.2501.13088,
title = {Every 2-quasitrace is a trace},
author = {Alec Gow},
journal= {arXiv preprint arXiv:2501.13088},
year = {2025}
}
Comments
There is a critical conflation between a W*-algebra and a concrete von Neumann algebra in Proposition 5.2. A priori, there is no reason that each of the W*-algebras of Proposition 5.2 should have a von Neumann algebra representation on the same Hilbert space, which is required in Pedersen's Theorem (Theorem 4.1). This leaves a critical gap in the argument of Theorem A (Corollary 5.3)