I know von Neumann algebras are generated by their projections. The proofs I've seen use spectral measures (this is the case in Murphy's and Conway's texts). I was wondering if there's a more elementary proof avoiding the machinery of spectral measure, perhaps with just some clever usage of the double commutant theorem?
I'd appreciate either a proof or a reference to one.
I would say you cannot expect something like that. Consider for instance a masa in $B(H)$: its equal to its own commutant, so any information from the commutant is already in the algebra.
The general definition of von Neumann algebra doesn't give you a clue of what the elements are like, so it is not surprising that one needs tools to construct them, and that's where the spectral theorem appears. I would say that even in finite-dimension you need the spectral theorem (just easier to prove it) to show that finite-dimensional von Neumann algebras are matrix algebras (as the first step is to use the minimal central projections).
Finally, note that the spectral theorem gives you something that you couldn't possibly get from the double commutant theorem: that a von Neumann algebra is the norm-closure of the span of its projections. More puzzling (and less obvious), $B(H)$ is actually spanned by its projections.