On the Quasitrace Problem and a Characterization of W*-algebras
Alec Gow
TL;DR
The paper develops a space-free program to characterize W$^*$-algebras via MASAs, linking intrinsic MASA structure to fundamental open questions about quasitraces and Kaplansky’s conjecture. It shows that, in the finite case, the W$^*$-Pedersen characterization is equivalent to every $2$-quasitrace being a trace, and it establishes a network of implications connecting finite MASA-W$^*$-properties, Kaplansky’s conjecture, and the $2$-quasitrace conjecture (2Q). Haagerup’s metric on quasitraces and the completion arguments for MASAs on Type II$_1$ factors are central to these results, while partial progress is obtained for countably decomposable properly infinite AW$^*$-factors. The work also sketches a pathway to relate (quasi)linearity of functionals on AW$^*$-algebras to monotone completeness, highlighting deep structural questions about AW$^*$-algebras and their biduals with potential impacts on operator algebra theory and classification.
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.