Let $A$ be a unital C\*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}\_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$


Suppose that $A$ satisfies the following property:


For every $S\subseteq A$ there is a projection $q\in A$ such that
$\operatorname{Ann}\_r(S)=Aq$.



> 
> Q. Is $A$ necessarily a von Neumann algebra?
> 
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> 



Impact of annihilators in C*-algebras
An AW${}^\*$-algebra is a C${}^\*$-algebra which satisfies this condition for both right and left annihilators. So every AW${}^\*$-algebra has your property, and any C${}^\*$ algebra that is isomorphic to its opposite algebra has your property iff it is AW${}^\*$.


There are lots of AW${}^\*$-algebras that aren't von Neumann algebras, even in the commutative case.


