# Origins of Stone duality

My question is a mix of mathematical and historical, if you consider my question will be better answered in the mathematics community, please tell me.
I want to know the historical roots of Stone's representation theorem. Here is what I have found:

• In Stanford Encyclopedia of Philosophy, it says that one of the roots and applications of the duality is in the "algebra of projections (Functional analysis)"
• In "The theory of representations for Boolean algebras" M.H.Stone says:

The writer's interest in the subject, for example, arose in connection with the spectral theory of symmetric transformations in Hilbert space and certain related properties of abstract integrals.

I have tried to look in "Linear transformations in Hilbert Space and their application to analysis" by M.H.Stone, in the chapter IX "Symmetric transformations" but my knowledge in functional analysis is not very advanced and I'm not used to the old notation that is used in it.
What I'm trying to find is where do the boolean algebras and booleans rings appear in functional analysis.
Again, if you think could be better answered in mathematics section, i would be grateful if someone could tell how to redirect it.

• One thing to try is look through Stone's earlier publications (many are freely available AMS publications), and don't overlook his abstracts in the Bull. AMS various "Abstracts of Papers" (e.g. see Abstract #86 on p. 200 here). This 1934 Proc. Nat. Acad. Sci. paper is an early (and freely available) paper, and this 1938 paper is a useful survey. – Dave L Renfro Apr 11 at 16:44
• I think the technical relations between Hilbert space spectral theory and Stone spaces is more suitable for MathOverflow. Hermitian projection operators that commute with a given self-adjoint operator (or a family thereof) form a Boolean algebra. It plays a key role in the spectral representation of the operator in terms of projection-valued measures and the related functional calculus. – Conifold Apr 12 at 5:17
• @DaveLRenfro Thank you, the last reference to a 1938 survey got my closer to what i want. – Iván Jorro Medina Apr 12 at 12:56
• @Conifold Thank you, very much of what you said was what I'm looking for. Also how can I redirect this without having to do copy paste? Can I do that? – Iván Jorro Medina Apr 12 at 18:16
• I would to save both comments of yours as they both get me closer to what i want to know, so if I have to make copy paste to do it, let me know. – Iván Jorro Medina Apr 12 at 18:18