I know the following result is true for a von Neumann algebra $A$:
Theorem: Let $A$ be a von Neumann algebra. Then every self-adjoint element $h\in A$, i.e $h=h^*$, is the limit of a sequence of linear combinations of mutually orthogonal idempotent elements of $A$.
I am, however, looking for a textbook in which this result is mentioned and a proof is given.
Can anyone please help provide me with such a reference?
The book A Course in Functional Analysis by John Conway (the analyst, not the group theorist) contains all the necessary ingredients in Chapter IX.
Suppose the von Neumann algebra acts on the Hilbert space $H$, and $h\in A$ is self-adjoint. Using theorems IX$.2.2$ (the spectral theorem) and theorem IX$.1.10$, $h$ is the limit of a sequence of linear combinations of orthogonal projections in $\mathcal B(H)$ (the projections come from the spectral measure associated with $h$ on disjoint subsets of $\sigma(h)$). By Theorem IX$.2.2(c)$ and Fuglede's theorem (a specific case of Theorem IX$.6.7$), all of these projections are in $A''$, and therefore in $A$.