The W$^*$-Pedersen characterization by maximal abelian self-adjoint subalgebras

Let $\mathcal{A}$ be a unital C$^*$-algebra. A MASA is a maximal abelian C$^*$-subalgebra of $\mathcal{A}$, and a W$^*$-algebra is a C$^*$-algebra admitting a Banach-space predual.

W$^*$-Pedersen characterization. The algebra $\mathcal{A}$ is a W$^*$-algebra if and only if every MASA of $\mathcal{A}$ is a W$^*$-algebra.

This is proposed as an intrinsic, space-free analogue of Pedersen's characterization. The source indicates that the characterization is established in certain finite cases conditional on the equivalent Kaplansky and $2$-quastrace assertions, while the properly infinite case remains conjectural.