Every 2-quasitrace is a trace

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.

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)