# Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras:

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.

• Just in case somebody stumbles upon this in the future, I will add also a link to the MO copy of the question: Masas in SAW*-algebras. – Martin Sleziak Dec 13 '17 at 13:51