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

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.