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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?

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    $\begingroup$ The definition you are looking for is "real rank 0"(RR0). It was introduced by L.Brown and Pedersen (JFA 1991)--see Theorem 2.6 especially as it pertains to your question. The class of (non von Neumann) RR0 C*-algebras is rich. As a first class of examples, all AF algebras (inductive limits of finite dimensional C*-algebras) have RR0 (this is easy to see directly since the class of algebras you wish to describe is clearly closed under injective inductive limits). If you're interested in more examples you cansearch for "simple AT algebras" which will probably lead you to even more examples. $\endgroup$
    – Caleb Eckhardt
    Commented Sep 21, 2017 at 0:00

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Isn't $c_0$ already a counterexample?

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  • $\begingroup$ Thank you very much. I can see that this is a $C^*$-algebra that is not von Neumann (it does not contain the identity of $c$). However, I am not able to see how every self-adjoint element is the limit of a sequence of linear combinations of orthogonal idempotents. Can you please elaborate on that? $\endgroup$
    – user860374
    Commented Oct 3, 2017 at 9:27
  • $\begingroup$ Every element of $c_0$ is the limit of a sequence of linear combinations of the elements of the standard basis. $\endgroup$
    – Nik Weaver
    Commented Oct 3, 2017 at 11:46
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Every monotone closed $C^*$-algebras has this property. And they are not all von-Neumann algebras. Although the counter-exemple are rather enormous objects.

In fact as commutative sub-algebra of $AW^*$ algebras are monotone closed, every $AW^*$-algebra have this property, but it is still open whether or not every $AW^*$ algebra is monotone closed.

A little bit more generally, it seems that a (unital) commutative $C^*$-algebra has this property if and only if its spectrum is a stone space (which is a little bit more general than monotone complete $C^*$-algebras which corresponds to Stonean space, and make it a lot easier to find non Von Neumann examples).

If you want to include the non-unital case as well, I'm extending the meaning of stone space to include spectrum of 'non-unital boolean algebra' i.e. locally compact topological space with a basis of compact clopen sets.

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