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Inspired by this question we ask the following question:

Is there a complete classification of all unital $C^*$ algebra $A$ for which the following subalgebra $B$ is a von Neumann algebra? Is there a terminology for such kind of $C^*$ algebras?

$$B=\text{The unital $C^*$ sub algebra generated by all projections of $A$} $$

Of course every von neumann algebra satisfies this property.

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No, not at all. Infinite-dimensional von Neumann algebras are non-separable as C$^*$-algebras, so for separable C$^*$-algebras (the ones one cares about) only trivial examples (i.e., finite-dimensional) are available. That will happen for instance when $A$ is projectionless and unital, where $B=\mathbb C$.

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