Every 2-quasitrace is a trace

TL;DR

This paper proves that every Type II_1 AW*-factor is a von Neumann algebra, and consequently, every 2-quasitrace on a unital C*-algebra is a trace, resolving longstanding open questions.

Contribution

It establishes that all Type II_1 AW*-factors are von Neumann algebras and confirms that 2-quasitraces are traces on unital C*-algebras, answering open problems.

Findings

Every Type II_1 AW*-factor is a von Neumann algebra.

All 2-quasitraces on unital C*-algebras are traces.

Resolved longstanding open questions in operator algebra theory.

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.