Analytical origins of the Stone duality I've asked this question in the HSM community, but by the nature of my question, some user told me to ask this question here.
This is the original post https://hsm.stackexchange.com/q/13087/14296

What I want to know is how M.H.Stone, by his research in functional analysis came to the Stone duality.
My knowledge of funtional analysis is not very advanced, what I have done is only a semester of bounded operators, that's my formation in it, so please if you can explain your answer a little bit, it would be great.
What I have been told (in the cited post) is that:

Hermitian projection operators that commute with a given self-adjoint operator (or a family thereof) form a Boolean algebra

Also that:

It plays a key role in the spectral representation of the operator in terms of projection-valued measures and the related functional calculus

What I want to also know is the technical relations between Hilbert space spectral theory and Stone spaces.
If you consider my tags to be wrong, please tell me. Also if you consider that this post, should be in Mathematics Stack-Exchange.
 A: Stone himself gave a brief account of his discovery of Stone duality in a letter written in 1976.
Apparently, von Neumann was involved.
A reminiscence on the extension of the Weierstrass approximation theorem
A Letter from Marshall Stone
Arnold House, Univ. of Massachusetts, Amherst
(The following is an excerpt from a letter to Israel Halperin of 16 February 1976, which was stimulated by a conversation on the origins of the Stone-Weierstrass theorem.)

“So far as I can reconstruct events, they went something like this.
In studying ways of constructing or representing topological spaces, as I did using Boolean algebra techniques with filters or ideals, I wanted to specialize to various familiar types of space--e.g. completely regular spaces (characterized by the presence of ‘adequate’ families of real continuous functions).
A well known theorem of Banach about metric spaces suggested that the algebraic structure of the bounded continuous function-ring should give deep information about the underlying space.
In trying to prove that such was the case, a logical analysis of various situations led directly to a need for a generalized Weierstrass approximation theorem.
As I recall this occurred at an early stage, where the meaning of ring isomorphism was at issue.
There was a moment when homomorphisms had to be taken up.
The stimulus came from Von Neumann.
He was visiting me in Cambridge.
We were walking across the Cambridge Common, and as I walked I was describing some of my work in this field.
He at once asked if it would be possible (in the case of compact spaces at least) to correlate ring homomorphisms with continuous maps of the underlying spaces.
The next day I gave him the affirmative response that I later published as part of my general theory.”


“Perhaps this and other things that have happened in the course of my research suggest that in many kinds of mathematical work the key is asking the ‘right’ questions.
Once the question is posed the answer becomes a matter of persistent analysis.
Of course, the big ‘unsolved’ problems (Fermat theorem, Riemann hypothesis, etc.) may provide counterexamples.
Still many problems seem to become easier when they can be twisted somehow into new forms converting them into ‘right’ questions.”

A: In addition to the historical component, the question also asked about the relation between spectral theory and Stone spaces.
$\def\Z{{\bf Z}} \def\R{{\bf R}} \def\C{{\bf C}} \def\Spec{\mathop{\rm Spec}} \def\Proj{\mathop{\rm Proj}} \def\Cont{{\rm C}} \def\MSpec{\mathop{\rm MSpec}} \def\Li{{\rm L}^∞}$
Given an algebra-like object $A$, we assign to it its poset of ideals
(typically defined as kernels of homomorphisms $A→B$),
which is interpreted as the poset of opens of some space $S$.
The technical term for such posets is locale,
which is a notion very closely related to topological spaces.
In particular, from any locale one can canonically extract a topological space,
and this is the topological space $S$ produced in many classical Stone-type dualities.
The points of $S$ are ideals corresponding to morphisms $A→k$, where $k$ is often a particularly simple algebra.
These often turn out to be maximal ideals in $A$.
Conversely, given a space-like object $S$, we assign to it the algebra of morphisms $S→k$,
where $k$ is often the “same” algebra $k$ as above,
only this time its underlying object is a space, not just a set.
Some examples from general topology, measure theory, differential geometry, algebraic geometry, and complex geometry
(the list is very much incomplete):




algebra
homomorphism
$k$
ideal
space
maps




Boolean algebra
homomorphism
$\Z/2$
ideal
compact totally disconnected Hausdorff
continuous map


complete Boolean algebra
complete homomorphism
$\Z/2$
closed ideal
compact extremally disconnected Hausdorff
open continuous map


localizable Boolean algebra
complete homomorphism
$\Z/2$
closed ideal
hyperstonean space
open continuous map


localizable Boolean algebra
complete homomorphism
$\Z/2$
closed ideal
compact strictly localizable enhanced measurable space
measurable map


commutative von Neumann algebra
normal *-homomorphism
$\C$
closed *-ideal
compact strictly localizable enhanced measurable space
measurable map


commutative unital C*-algebra
*-homomorphism
$\C$
closed *-ideal
compact Hausdorff space
continuous map


commutative algebra over $k$
homomorphism
$k$
ideal
coherent space / affine scheme
continuous map / morphism of schemes


finitely generated germ-determined C$^∞$-ring
C$^∞$-homomorphism
$\R$
germ-determined ideal
smooth locus (e.g., smooth manifold)
smooth map


finitely presented complex EFC-algebra
EFC-homomorphism
$\C$
ideal
globally finitely presented Stein space
holomorphic map




The duality relevant to the spectral theory is the duality between commutative von Neumann algebras
and compact strictly localizable enhanced measurable spaces.
Given a normal operator $T$ on a Hilbert space $H$,
$T$ generates a commutative von Neumann algebra $A$ inside $B(H)$, i.e., bounded operators on $H$.
(This is precisely the point where normality is crucial; without the relation $T^*T=TT^*$
the algebra generated by $T$ will be noncommutative.)
More concretely, $A$ can be described as the closure in the ultraweak
topology on $B(H)$ of the set of polynomials with complex coefficients
in variables $T$ and $T^*$, i.e., finite sums $∑_{i,j}a_{i,j}T^i(T^*)^j$ for arbitrary $a_{i,j}∈{\bf C}$.
By the cited duality, the commutative von Neumann algebra $A$
is dual to a compact strictly localizable enhanced measurable space $\Spec A$.
This is indeed the spectrum of $T$ in the usual sense.
Under this equivalence, the element $T∈A$
corresponds to the measurable map $\Spec A→\C$ given by inclusion of $\Spec A$ into $\C$.
More concretely, we can extract all projections from $A$ (defined as self-adjoint idempotents, $T^2=T$ and $T^*=T$),
and these form a complete Boolean algebra $\Proj A$.
The easiest way to see this is to observe that Boolean algebras
are precisely rings in which $x^2=x$.
The ring identities are trivially satisfied by definition of projections.
Completeness is implied by the fact that $A$ is closed
in the ultraweak topology.
The Stone duality converts the Boolean algebra $\Proj A$
into a topological space, its Stone spectrum $\Spec A$.
Points in the Stone spectrum are precisely the maximal
ideals $P$ in the Boolean algebra $\Proj A$.
Open sets correspond to ideals $I$ of $A$:
a point $P$ belongs to the open set corresponding to $I$
if $P$ does not contain $I$ as an ideal.
In fact, the topological space $\Spec A$
also coincides with the Gelfand spectrum
of $A$, interpreted as a commutative unital C-algebra;
so points in $\Spec A$ can also be interpreted as maximal
closed -ideals $I$ in $A$ as a commutative C-algebra,
or, equivalently, as homomorphisms of unital C-algebras $A→\C$.
Furthermore, we have a canonical isomorphism of von Neumann algebras
$$R\colon A→\Cont(\Spec A,\C),$$
where $\Cont(\Spec A,\C)$ denotes the algebra of continuous complex-valued functions on $\Spec A$.
Concretely, given an element $a∈A$ and a point $p∈\Spec A$
given by the homomorphism $A→\C$ of C*-algebras,
we set $T(a)(p)=p(a)$.
This is the Gelfand transform of $a$.
The topological space $\Spec A$ belongs to a very special
class of topological spaces, the hyperstonean spaces.
From such a space one can extract a σ-algebra $M$ of measurable
sets and a σ-ideal $N$ of negligible sets (alias sets of measure 0),
both on the set $\Spec A$.
The resulting triple $\MSpec A=(\Spec A,M,N)$ is an example of
an enhanced measurable space.
The elements of $N$ are precisely the nowhere dense subsets of $\Spec A$,
which coincides with meager subsets (alias sets of first category).
The elements of $M$ are precisely the symmetric differences
of clopen (closed and open) subsets of $\Spec A$ and elements of $N$.
(This is sometimes referred to as the Loomis–Sikorski construction
for $\Spec A$; see John C. Oxtoby's book Measure and Category for more on this topic.)
Continuing the above line of reasoning,
we have a canonical isomorphism of von Neumann algebras
$$S\colon A→\Li(\MSpec A,\C),$$
where $\Li(\MSpec A,\C)$ denotes the algebra of
equivalence classes of complex-valued measurable functions
$\MSpec A→\C$ modulo the equivalence relation of equality
on a conegligible set (alias equality almost everywhere).
(Indeed, one can prove that $\Cont(\Spec A,\C)$
is canonically isomorphic to $\Li(\MSpec A,\C)$.)
Under this correspondence, the original operator $T∈B(H)$
corresponds to an element $S(T)∈\Li(\MSpec A,\C)$.
The map $R(T)\colon \Spec A→\C$ allows us to interpret ponts of $\Spec A$
(and therefore also of $\MSpec A$)
as complex numbers.
This identifies $\MSpec A$ with the usual spectrum of the operator $T∈B(H)$.
Furthermore, the isomorphism
$$S\colon A→\Li(\MSpec A,\C),$$
is known as the Borel functional calculus of $T∈B(H)$.
More precisely, given a bounded Borel-measurable function $f\colon \C\to\C$, the element $S^{-1}(f\circ R(T))∈A⊂B(H)$
is precisely the operator $f(T)∈B(H)$ given
by the traditional Borel functional calculus.
One may ask whether we can recover the full spectral theorem for a normal operator in this manner.
This is possible once Stone duality is upgraded to Serre–Swan-type duality
between modules and vector bundle-like objects (including, e.g., sheaves etc.).
Given a vector bundle-like object $V→S$, we assign to it its module of sections,
which is a module over the algebra of maps $S→k$.
Conversely, given a module $M$ over $A$,
the corresponding vector bundle-like object $V→S$ over $S=\Spec A$
has as its fiber over some point $s∈S$ the vector space $M/IM$,
where $I$ is the ideal corresponding to $s$.
(Many details are necessarily omitted in this brief sketch.)
Typically, genuine vector bundles correspond to dualizable modules
(dualizable with respect to the tensor product over $A$).
Non-dualizable module tend to correspond to sheaves that are not vector bundles,
e.g., skyscraper sheaves etc.




module
vector-bundle-like object




module over a Boolean algebra
sheaf of $\Z/2$-vector spaces


Hilbert W*-module over a commutative von Neumann algebra
measurable field of Hilbert spaces


representations of a commutative von Neumann algebra on a Hilbert space
measurable field of Hilbert spaces


Hilbert C*-module over a commutative unital C*-algebra
continuous field of Hilbert spaces


module over a commutative algebra over $k$
sheaf of modules over an affine scheme


dualizable module over a commutative algebra over $k$
algebraic vector bundle


dualizable module over a finitely generated germ-determined C$^∞$-ring
smooth vector bundle


dualizable module over finitely presented complex EFC-algebra
holomorphic vector bundle




The duality relevant to the spectral theory is the duality between representations of a commutative von Neumann algebras on a Hilbert space
and measurable fields of Hilbert spaces.
Given a normal operator $T$ on a Hilbert space $H$,
$T$ generates a commutative von Neumann algebra $A$ inside $B(H)$,
whose spectrum $\Spec A$ is a compact strictly localizable enhanced measurable space.
Furthermore, the inclusion of $A$ into $B(H)$ is a representation of $A$ on $H$.
As such, it corresponds under the Serre–Swan-type duality to a measurable field of Hilbert spaces over $A$.
This is precisely the measurable field produced by the classical spectral theorem.
Under the duality, the operator $T$ corresponds to the operator that
multiplies a given section of this measurable field of Hilbert spaces
by the complex-valued function $\Spec A→\C$ produced above.
Thus, we recovered the entire content of the classical spectral theorem.
In fact, the above considerations work equally well to establish the spectral
theorem for an arbitrary family (not necessarily finite) of commuting normal operators.
See also the nLab article duality between algebra and geometry, which may contain additional updates.
