Any commutative AW$^*$-algebra has the form $C(X)$ where $X$ is a Stonean space.
In Takesaki's book Theory of Operator Algebra's, he shows that any Stonean space can be decomposed into three parts: a hyperstonean part, a part with a dense meagre set and a part where every regular positive measure has a nowhere dense support. Let's call these Type A,B and C.
Type A is a von Neumann algebra, while type B and C allow no normal states, so they are in a certain sense maximally non-von Neumann.
A Type C Stonean space doesn't allow a faithful state, since all states/measures have nowhere dense support.
My question is: Is there a Type B commutative AW$^*$-algebra that has a faithful state? Or in other words: Does there exist a Stonean space with a dense meagre set that has a measure on it that is nonzero on every open set.
edit: Just want to add that the question can be reframed in another way: Is there a commutative AW$^*$-algebra with a faithful state that isn't a von Neumann algebra? There is a result that says that any AW$^*$ II-factor with a faithful state is a von Neumann algebra, while there are III-factor AW$^*$-algebra's that aren't von Neumann, so I wouldn't be surprised if there was some result about the commutative case in either direction.