Consider the following theorem by Aupetit.

Let $A$ and $B$ be two von-Neumann algebras and let $\phi$ be a spectrum-preserving linear mapping from $A$ onto $B$. Then $\phi$ is a Jordan isomorphism.

The statement can be relaxed further by allowing $B$ be to be any semi-simple Banach algebra without altering the proof by Aupetit. However, I am thefefore lead to wondering if the von Neumann assumption on $A$ can also be relaxed. Here is my trail of thought:

The only place in the proof of Aupetit where we use the assumption that $A$ is von-Neumann, is where we invoke the fact that every self-adjoint element in $A$ is the limit of a sequence of linear combinations of orthogonal idempotents.

Could this theorem by Aupetit therefore not get generalized to:

Let $A$ be a $C^*$-algebra such that every self-adjoint elements is the limit of a sequence of linear combinations of orthogonal idempotents and $B$ be a semi-simple Banach algebra. If $\phi$ is a spectrum-preserving linear mapping from $A$ onto $B$, then $\phi$ is a Jordan isomorphism.

Based on the proof by Aupetit, I am lead to believe that the answer is yes, but whether this result is in fact more general than the one given by Aupetit, relies on the following question:

Is there a $C^*$-algebra which is not a von Neumann algebra, which satisfies the property that every self-ajoint element is the limit of a sequence of linear combinations of orthogonal idempotents.

This I am not sure about. Can anyone perhaps point me in the correct direction as to finding such an example?