nLab
Gelfand duality

Contents

Context

Algebra

Geometry

Functional analysis

Duality

Noncommutative geometry

Contents

Idea

Gelfand duality is a duality between spaces and their algebras of functions for the case of compact topological spaces and commutative C-star algebras:

Every (nonunital) C *C^\ast-algebra AA is equivalent to the C *C^\ast-algebra of continuous functions on the topological space called its Gelfand spectrum sp(A)sp(A).

This theorem is the basis for regarding non-commutative C *C^\ast-algebras as formal duals to spaces in noncommutative geometry.

Definitions

The statement of Gelfand duality involves the following categories and functors.

Definition

Write

  • C *AlgC^\ast Alg for the category of unital C-star algebras;

  • C *Alg nuC^\ast Alg_{nu} for the category of nonunital C *C^\ast-algebras;

  • C *Alg nu,ndC^\ast Alg_{nu,nd} for the sub category of nonunital C *C^\ast-algebras with morphisms being nondegenerate -homomorphisms?, i.e., the two-sided ideal generated by the image is dense in the codomain.

  • C *Alg comC^\ast Alg_{com}, C *Alg com,nuC^\ast Alg_{com,nu}, C *Alg com,nu,ndC^\ast Alg_{com,nu,nd} for the full subcategories of the above three categories consisting of commutative C*-algebras.

And

The duality itself is exhibited by the following functors:

Definition

Write

C:Top cptC *Alg com op C \;\colon\; Top_{cpt} \to C^\ast Alg_{com}^{op}

for the functor which sends a compact topological space XX to the algebra of continuous functions C(X)={f:X|fcontinuous}C(X) = \{f : X \to \mathbb{C} | f \; continuous\}, equipped with the structure of a C *C^\ast-algebra in the evident way (…).

Write

C 0:*/Top cptC *Alg com,nu C_0 \;\colon\; */Top_{cpt} \to C^\ast Alg_{com,nu}

for the functor that sends a pointed topological compact Hausdorff space (X,x 0)(X,x_0) to the algebra of continuous functions f:Xf : X \to \mathbb{C} for which f(x 0)=0f(x_0) = 0.

Definition

Write

sp:C *Alg com opTop cpt sp : C^\ast Alg_{com}^{op} \to Top_{cpt}

for the Gelfand spectrum functor: it sends a commutative C *C^\ast-algebra AA to the set of characters – non-vanishing C *C^\ast-algebra homomorphisms x:Ax : A \to \mathbb{C} – equipped with the spectral topology.

Similarly write

sp:C *Alg com,nu opTop lcpt. sp : C^\ast Alg_{com,nu}^{op} \to Top_{lcpt} \,.

Statement

Theorem

(Unital Gelfand duality theorem)

The pair of functors

C *Alg com opspCTop cpt C^\ast Alg_{com}^{op} \stackrel{\overset{C}{\leftarrow}}{\underset{sp}{\to}} Top_{cpt}

is an equivalence of categories.

Here C *Alg opC^\ast Alg^{op}_{\cdots} denotes the opposite category of C *Alg C^\ast Alg_{\cdots}.

Corollary

On non-unital C *C^\ast-algebras the above induces an equivalence of categories

C *Alg com,nu opspC 0*/Top cpt. C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} */Top_{cpt} \,.
Proof

The operation of unitalization () +(-)^+ constitutes an equivalence of categories

C *Alg nu() +kerC *Alg/ C^\ast Alg_{nu} \stackrel{\overset{ker}{\leftarrow}}{\underset{(-)^+}{\to}} C^\ast Alg / \mathbb{C}

between non-unital C *C^\ast-algebras and the over-category of C *C^\ast-algebras over the complex numbers \mathbb{C}.

Composed with the equivalence of theorem this yields

C *Alg com,nu op() +(C *Alg com/) opC*/Top cpt. C^\ast Alg_{com,nu}^{op} \underoverset{\simeq}{(-)^+}{\to} (C^\ast Alg_{com}/\mathbb{C})^{op} \underoverset{\simeq}{C}{\to} * / Top_{cpt} \,.

The weak inverse of this is the composite functor

C 0:*/Top cptsp(C *Alg com/) opkerC *Alg com,nu op C_0 : */Top_{cpt} \underoverset{\simeq}{sp}{\to} (C^\ast Alg_{com}/\mathbb{C})^{op} \underoverset{\simeq}{ker}{\to} C^\ast Alg_{com,nu}^{op}

which sends (*x 0X)(* \stackrel{x_0}{\to} X) to ker(C(X)ev x 0)ker(C(X) \stackrel{ev_{x_0}}{\to} \mathbb{C}), hence to {fC(X)|f(x 0)=0}\{f \in C(X) | f(x_0) = 0\}. This is indeed C 0C_0 from def. .

Remark

Since locally compact Hausdorff spaces are equivalently open subspaces of compact Hausdorff spaces, via the construction that sends a locally compact Hausdorff space XX to its one-point compactification, and since a continuous function on the compact Hausdorff space X *X^\ast which vanishes at the extra point is equivalently a continuous function on XX which vanishes at infinity, the above induces a contravariant equivalence

Top lcpt,infC *Alg com,nuTop_{lcpt,inf} \leftrightarrows C^\ast Alg_{com,nu}

between the category of locally compact Hausdorff spaces and continuous maps vanishing at infinity and the category of commutative nonunital C*-algebras.

The duality also works with real numbers instead of complex numbers (Johnstone 82, chapter IV)

For an overview of other generalizations see also this MO discussion.

If one uses nondegenerate morphisms of C*-algebras instead, the duality works for [locally compact topological spaces]] and proper maps. See for instance (Brandenburg 07).

Theorem

The is an equivalence of categories

Top lcpt,properC *Alg com,nu,ndTop_{lcpt,proper} \leftrightarrows C^\ast Alg_{com,nu,nd}

between the category of locally compact Hausdorff spaces and proper maps and the category of commutative nonunital C*-algebras with nondegenerate *-homomorphisms as morphisms.

Generalizations

In constructive mathematics

Gelfand duality makes sense in constructive mathematics hence internal to any topos: see constructive Gelfand duality theorem.

By horizontal categorification

Gelfand duality can be extended by horizontal categorification to define the notion of spaceoids as formal duals of commutative C *C^*-categories.

The analogous statement in differential geometry:

Isbell duality between algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}geometryA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\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 }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A}superalgebra
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}algebraA\phantom{A}A\phantom{A}geometryA\phantom{A}
A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}AQFTA\phantom{A}A\phantom{A}FQFTA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

References

Formulation of Gelfand duality in terms of category theory (adjoint functors, monads (“triples”) and adjoint equivalences) originates with

  • Joan W. Negrepontis, Duality in analysis from the point of view of triples, Journal of Algebra, 19 (2): 228–253, (1971) (doi, ISSN 0021-8693, MR 0280571)

Quick exposition is in

  • Ivo Dell’Ambrogio, Categories of C *C^\ast-algebras (pdf)

Textbook accounts include

  • Peter Johnstone, section IV.4 of Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press 1982. xxi+370 pp. MR85f:54002, reprinted 1986.

  • N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics 1998. xx+529 pp. MR2000g:81081 doi

  • Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. gBooks

Careful discussion of the duality for the more general case of locally compact topological spaces includes

Discussion of Gelfand duality as a fixed point equivalence of an adjunction between includes

following

Some other generalized contexts for Gelfand duality:

Last revised on May 22, 2020 at 16:47:15. See the history of this page for a list of all contributions to it.