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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.

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    $\begingroup$ I don't have it on hand to check but I believe the book Stone spaces has some useful historical information. $\endgroup$ – Noah Schweber Apr 13 at 12:48
  • $\begingroup$ @NoahSchweber I have checked the book you mention, and it explains like timeline of this representation theorem, but i want to know some of the functional analysis details of how he got there. $\endgroup$ – Iván Jorro Medina Apr 13 at 15:55
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    $\begingroup$ “Hermitian projection operators that commute with a given self-adjoint operator (or a family thereof) form a Boolean algebra”: This statement is false. The operator id∈B(H) is self-adjoint and any projection in B(H) commutes with id. But projections in B(H) do not form a Boolean algebra. Projections in the von Neumann algebra generated by a given self-adjoint operator do form a complete Boolean algebra. $\endgroup$ – Dmitri Pavlov Apr 17 at 16:07
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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.

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  • $\begingroup$ I'm sorry, but if I say I have understand a bit of what is being said, I would be lying. To only explain it so I can get the big picture, I would need to ask too many questions. So I decided that I am going to leave this right here, so in the future I learn more category theory, functional analysis and more math in general, as I am still an undergraduate. $\endgroup$ – Iván Jorro Medina Apr 14 at 22:32
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    $\begingroup$ This may sound naive and maybe dumb, but this makes me look at very cool things and gives me motivation for studing more math, so thank you for your answer and for the indirect motivation. I did this posts only to give some historical and technical details about the origins of the duality for my final project, that's all, but I get much more than that, even if I can't understand it by now. I would like to thank you and all the people that take the time to at least think about it. I leave this posts to leave the door open for other answers. $\endgroup$ – Iván Jorro Medina Apr 14 at 22:41
  • $\begingroup$ @IvánJorroMedina: The motivation mentioned in the first sentence was roughly my motivation for writing the answer the way I wrote it (i.e., explaining the bigger picture instead of just one narrow aspect of it), so it is quite pleasing to see that I had some success with it. $\endgroup$ – Dmitri Pavlov Apr 15 at 18:14
  • $\begingroup$ Could you please, show the details of the things I mention in the two quotes without going to much into other areas and just staying only with functional analysis? It's the answer I want, I like this answer that you have already made, but it's too difficult for me to understand it. If this is too basic, I will be grateful if you could recomend some book, where I can find this, as I have said my knowledge of functional analysis is very narrow. $\endgroup$ – Iván Jorro Medina Apr 16 at 1:00
  • $\begingroup$ @IvánJorroMedina: Okay, I added a detailed explanation of how to recover the Borel functional calculus in this setting. The statement in the first quote is false, though, as I indicated in another comment. I added what I think was the intended meaning of that statement in my answer. My current favorite book on functional analysis is Helemskii's Lectures and Exercises on Functional Analysis (AMS, 2006). $\endgroup$ – Dmitri Pavlov Apr 17 at 16:11
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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.”

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    $\begingroup$ This note discusses his discovery of the Stone-Weierstrass theorem, not Stone duality. "As I did using Boolean algebra techniques" apparently refers to his having proved the duality result some time earlier. $\endgroup$ – Nik Weaver Apr 13 at 21:00
  • $\begingroup$ @NikWeaver: How do you suggest to interpret the phrase "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" as pertaining to something else than the Stone duality (which says that the two sets are indeed isomorphic)? $\endgroup$ – Dmitri Pavlov Apr 13 at 22:14
  • $\begingroup$ Thank you for the answer, it is very interesting and amazing how Von Neumann was present in some stories and has great insight, like the one with Gödel. By looking at your profiles, I see that both of you could probably explain the technical detail of this, I mean, what is the thing that forms a boolean algebra? I don't want to know about the representation theorem, I want to know what I mention up in the quotes. $\endgroup$ – Iván Jorro Medina Apr 13 at 23:39
  • $\begingroup$ @IvánJorroMedina: Sure, I posted another answer (was too long to post as an addendum here). $\endgroup$ – Dmitri Pavlov Apr 14 at 0:54
  • $\begingroup$ @DmitriPavlov Stone duality does not say that. "Stone duality" refers to the duality between Boolean algebras and totally disconnected compact Hausdorff spaces, while the passage you quote refers to the duality between compact Hausdorff spaces and abelian C*-algebras. It is not about Stone duality. $\endgroup$ – Nik Weaver Apr 14 at 3:23

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