On the Quasitrace Problem and a Characterization of W*-algebras

Abstract

We conjecture that a unital C*-algebra is a W*-algebra if and only if each of its maximal abelian self-adjoint subalgebras is a W*-algebra; this is a space-free analogue of a known result due to G.K. Pedersen. Our main result is a proof that this conjecture holds for finite C*-algebras if and only if every $2$-quasitrace on a unital C*-algebra is a trace. We also show that the spatial condition in Pedersen's Theorem can be substantially weakened for AW*-factors. Finally, we give a new characterization of Type II${}_1$ W*-factors among Type II${}_1$ AW*-factors, which allows us to relate the question of (quasi)linearity of functionals on finite AW*-algebras to the question of monotone completeness of AW*-algebras.

Replaces arXiv:2501.13088 (withdrawn). v2: 40 pages. Significant updates to exposition and organization. New equivalence added to Theorem A. Theorem B updated. The Type II${}_{\infty }$ case now handled separately in Theorem C, thereby correcting an error from v1. Appendix removed due to v1 error; corrected results now contained in main body. Added Example 2.18, Section 3.1, and Section 4.4