# Classification of $C^*$ algebras whose subalgebra generated by projections is a von neumann algebra

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.

## 1 Answer

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 $$B=\mathbb C$$.