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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?

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    $\begingroup$ Do you mean "simple" (no nontrivial ideals) or "trivial centre"? They are not the same. $\endgroup$ – Martin Argerami Feb 14 '18 at 20:08
  • $\begingroup$ I meant trivial centre. I've been trying to find some results about the difference between simple and trivial centre. Could you maybe give me a reference for the difference (and if one implies the other). $\endgroup$ – John Feb 15 '18 at 17:18
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An easy counterexample, if the requirement is to have trivial centre, is to take $H$ infinite-dimensional, and put $A=K(H)+\mathbb C\,I$.

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