nLab Gelfand duality

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

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

Definitions

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

Definition

Write

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

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

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

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

And

• Top${}_{Haus}$ for the category of Hausdorff topological spaces

• $Top_{cpt}$ for the full subcategory of Top${}_{Haus}$ of the compact topological spaces;

• $*/Top_{cpt}$ for the category of pointed topological compact Hausdorff spaces, i.e. the pointed objects in $Top_{cpt}$;

• $Top_{lcpt,proper}$ for the category of Hausdorff and locally compact topological spaces with morphisms being the proper maps of topological spaces.

• $Top_{lcpt,inf}$ for the category of Hausdorff and locally compact topological spaces with morphisms being the continuous maps that vanish at infinity.

The duality itself is exhibited by the following functors:

Definition

Write

$$C \;\colon\; Top_{cpt} \to C^\ast Alg_{com}^{op}$$

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

Write

$$C_0 \;\colon\; */Top_{cpt} \to C^\ast Alg_{com,nu}^{op}$$

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

Definition

Write

$$sp : C^\ast Alg_{com}^{op} \to Top_{cpt}$$

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

Similarly write

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

Statement

Theorem

(Unital Gelfand duality theorem)

The pair of functors

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

is an equivalence of categories.

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

Corollary

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

$$C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} */Top_{cpt} \,.$$

Proof

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

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

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

Composed with the equivalence of theorem this yields

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

The weak inverse of this is the composite functor

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

which sends $(* \stackrel{x_0}{\to} X)$ to $ker(C(X) \stackrel{ev_{x_0}}{\to} \mathbb{C})$, hence to $\{f \in C(X) | f(x_0) = 0\}$. This is indeed $C_0$ 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 $X$ to its one-point compactification, and since a continuous function on the compact Hausdorff space $X^\ast$ which vanishes at the extra point is equivalently a continuous function on $X$ which vanishes at infinity, the above induces a contravariant equivalence

$$Top_{lcpt,inf} \leftrightarrows C^\ast Alg_{com,nu}$$

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

$$Top_{lcpt,proper} \leftrightarrows C^\ast Alg_{com,nu,nd}$$

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 $C^*$-categories.

Related concepts

• Isbell duality

  • Gelfand-Kolmogorov theorem

  • Gelfand duality, constructive Gelfand duality

    • C-star algebra, topological space

    • Gelfand-Naimark theorem

    • Gelfand spectrum

    • Serre-Swan theorem

    • noncommutative topology

• Stone duality

The analogous statement in differential geometry:

• embedding of smooth manifolds into formal duals of R-algebras

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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$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{<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}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\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)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$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$\phantom{A}$

References

The original reference is

• Israel Gelfand, Normierte Ringe, Recueil Mathématique 9(51):1 (1941), 3–24.

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 $C^\ast$-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 (1995) [pdf, gBooks]

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

• Martin Brandenburg, Gelfand-Dualität ohne 1, 2007 (web)

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

• Hans-E. Porst, Walter Tholen, section 4-c of Concrete Dualities in H. Herrlich, Hans-E. Porst (eds.) Category Theory at Work, Heldermann Verlag 1991 (pdf)

following

• Eduardo Dubuc, Horacio Porta, Convenient categories of topological algebras, Bull. Amer. Math. Soc., Volume 77, Number 6 (1971), 975-979 euclid

Some other generalized contexts for Gelfand duality:

• Hans Porst, Manfred B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta mathematica 25:2, (1978) pp 169-204

• Chris Heunen, Klaas Landsman, Bas Spitters, Sander Wolters, The Gelfand spectrum of a noncommutative $C^\ast$-algebra, J. Aust. Math. Soc. 90 (2011), 39–52 doi pdf

• Christopher J. Mulvey, A generalisation of Gelfand duality, J. Algebra 56, n. 2, (1979) 499–505 doi

• Arthur Parzygnat, Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem (arXiv:1708.00091)

category: analysis, geometry, noncommutative geometry