Reference request for c*-algebra embedding into the atomic part of its double dual

Question

Let $A$ be a C*-algebra. According to operator algebraists, it is well known that $A$ embeds into the atomic part of its double dual in the following sense: if $z$ is the central projection in $A^{**}$ onto its atomic part, then the composition of the embedding $A \hookrightarrow A^{**}$ with multiplication by $z$ yields an embedding $A \hookrightarrow zA^{**}$.

I see how one can prove this, but I need this result in my paper which is not aimed at operator algebraists, so I would like to give a reference. However, I cannot find this in the literature. I looked at Takesaki, Blackadar, Dixmier, Sakai, Kadison-Ringrose, and the closest I got was Definition III.6.35 in Takesaki where he defines the universal atomic representation, but I don't see how the above result immediately follows from this definition or its subsequent results.

Does anyone have a reference for this?

Answer 1

If you are only interested in seeing the statement written down in the literature, then see the first paragraph of the paper A Gelfand representation theory for C$^*$-algebras by C. Akemann. This in turn refers to Dixmier's text Les C$^*$-algebres et leurs représentations, page 39, but when I look at the results on that page in the text (about existence of representations of C*-algebras), it would not be apparent how to derive the claim unless one is already familiar with the theory of C$^*$-algebras.

I also just stumbled (via Akemann's earlier paper The general Stone-Weierstrass problem) upon the following:

On the Borel structure of C-algebras* by E.B. Davies

Theorem 3.2 of the paper is just a bit more general than what you are asking for, showing that a certain envelope of each C$^*$-algebra embeds in the reduced atomic representation.