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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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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.

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