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?



Reference request for c*-algebra embedding into the atomic part of its double dual
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](http://dx.doi.org/10.2140/pjm.1971.39.1)* 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](https://doi.org/10.1016/0022-1236(69)90015-9)*) upon the following: 


*[On the Borel structure of C](https://projecteuclid.org/euclid.cmp/1103840557)*-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.


