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

TL;DR

This paper explores a characterization of W*-algebras through maximal abelian subalgebras, proving it for finite C*-algebras if and only if all 2-quasitraces are traces, and discusses related structural properties.

Contribution

It establishes a new characterization of W*-algebras based on maximal abelian subalgebras and connects quasitraces to traces in finite C*-algebras.

Findings

Characterization holds for finite C*-algebras if all 2-quasitraces are traces.

Weakens conditions in Pedersen's theorem for countably decomposable AW*-factors.

Relates (quasi)linearity of functionals to monotone completeness of AW*-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 characterization 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 condition in (the spatial version of) Pedersen's Theorem can be substantially weakened in the case of countably decomposable AW$^*$-factors. We conclude with a preliminary result that allows us to relate the question of (quasi)linearity of functionals on AW$^*$-algebras to the question of monotone completeness of AW$^*$-algebras.