Let $A$ be a unital C$^*$-algebra such that the linear span of its projections lies norm dense, and such that below each projection there is an atomic projection. Suppose furthermore that $A$ is simple, i.e. that it has no non-trivial central elements.
Does $A$ have to be a von Neumann algebra? Since if this is true $A$ will be a type I von Neumann factor, the question reduces to: is $A$ isomorphic to $B(H)$ for some Hilbert space $H$?
A result by Kaplansky establishes this when $A$ is an AW$^*$-algebra, i.e. when its abelian subalgebras are monotone complete. Because the projections lie norm-dense the abelian subalgebras will be $\sigma$-monotone complete. Is this enough?