I asked this question three years ago at MSe but it has no response; let me try here.
Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, 15, No. 1 (1986), 15-32.):
A C*-algebra $A$ is an SAW-algebra* if for each pair of orthogonal, positive elements $x,y\in A$, there exists a positive element $e\in A$ such that $ex=x$ and $(1-e)y=y$.
This looks very commutatively, hence my question:
Let $B$ be a maximal abelian subalgebra of an SAW*-algebra. Is $B$ an SAW*-algebra?
This is the case for AW* algebras which are SAW* a fotriori.