# nLab Gelfand duality

Contents

### Context

#### Algebra

higher algebra

universal algebra

duality

## In QFT and String theory

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# 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) $C^\ast$-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

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

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

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

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

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)

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.

The analogous statement in differential geometry:

Isbell 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}$

## 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, 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:

• 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)

Last revised on March 3, 2021 at 05:03:00. See the history of this page for a list of all contributions to it.