# nLab duality between algebra and geometry

Contents

### Context

#### Algebra

higher algebra

universal algebra

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

In mathematics, one often has an equivalence of categories between algebra-like objects and space-like objects. Such an idea has many incarnations: Stone duality, Gelfand duality, etc., and in this article we make some observations that are common to these dualities.

Given an algebra-like object $A$, we assign to it its poset of ideals (typically defined as kernels of homomorphisms $A\to B$), 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 $A\to k$, 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 $S\to k$, 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):

algebrahomomorphism$k$idealspacemapsduality
Boolean algebrahomomorphism$\mathbf{Z}/2$idealcompact totally disconnected Hausdorff space (Stone space)continuous mapStone duality
complete Boolean algebracomplete homomorphism$\mathbf{Z}/2$closed idealcompact extremally disconnected Hausdorff space (Stonean space)open continuous mapStonean duality
localizable Boolean algebracomplete homomorphism$\mathbf{Z}/2$closed idealhyperstonean spaceopen continuous map
localizable Boolean algebracomplete homomorphism$\mathbf{Z}/2$closed idealcompact strictly localizable enhanced measurable spacemeasurable map
commutative von Neumann algebranormal *-homomorphism$\mathbf{C}$ultraweakly closed *-idealcompact strictly localizable enhanced measurable spacemeasurable map
commutative unital C*-algebra*-homomorphism$\mathbf{C}$norm-closed *-idealcompact Hausdorff spacecontinuous mapGelfand duality
commutative algebra over $k$homomorphism$k$idealcoherent space / affine schemecontinuous map / morphism of schemesZariski duality
finitely generated germ-determined C^∞-ringC$^\infty$-homomorphism$\mathbf{R}$germ-determined idealsmooth locus (e.g., smooth manifold)smooth mapMilnor duality
finitely presented complex EFC-algebraEFC-homomorphism$\mathbf{C}$idealglobally finitely presented Stein spaceholomorphic mapStein duality

and:

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

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.)

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 $T\in A$ corresponds to the measurable map $Spec A\to \mathbf{C}$ given by inclusion of $Spec A$ into $\mathbf{C}$.

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 of modules etc.).

Given a vector bundle-like object $V\to S$, we assign to it its module of sections, which is a module over the algebra of maps $S\to k$. Conversely, given a module $M$ over $A$, the corresponding vector bundle-like object $V\to S$ over $S=Spec A$ has as its fiber over some point $s\in S$ 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 of modules that are not vector bundles, e.g., skyscraper sheaves etc.

modulevector bundle-like object
module over a Boolean algebrasheaf of $\mathbf{Z}/2$-vector spaces
Hilbert W*-module over a commutative von Neumann algebrameasurable field of Hilbert spaces
W*-representations of a commutative von Neumann algebra on a Hilbert spacemeasurable field of Hilbert spaces
Hilbert C*-module over a commutative unital C*-algebracontinuous field of Hilbert spaces
module over a commutative ringquasicoherent sheaf of $\mathcal{O}$-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$^\infty$-ringsmooth vector bundle
dualizable module over finitely presented complex EFC-algebraholomorphic 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 $Spec A\to \mathbf{C}$ 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.

Last revised on July 29, 2023 at 15:14:24. See the history of this page for a list of all contributions to it.