# nLab Gelfand duality

### Context

#### Algebra

higher algebra

universal algebra

duality

## In QFT and String theory

#### Noncommutative geometry

noncommutative geometry

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

# Contents

## Idea

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

every (nonunital) ${C}^{*}$-algebra $A$ is equivalent to the ${C}^{*}$-algebra of continuous functions on the topological space called its Gelfand spectrum $\mathrm{sp}\left(A\right)$.

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

## Definitions

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

###### Definition

Write

• ${C}^{*}\mathrm{Alg}$ for the category of C-star algebras;

• ${C}^{*}{\mathrm{Alg}}_{\mathrm{nu}}$ for the category of non-unital ${C}^{*}$-algebras;

• ${C}^{*}{\mathrm{Alg}}_{\mathrm{com}}\subset {C}^{*}\mathrm{Alg}$ for the full subcategory of commutative ${C}^{*}$-algebras;

• ${C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}\subset {C}^{*}{\mathrm{Alg}}_{\mathrm{nu}}$ for the full subcategory of commutative non-unital ${C}^{*}$-algebras.

And

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

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

• $*/{\mathrm{Top}}_{\mathrm{cpt}}$ for the category of pointed objects in ${\mathrm{Top}}_{\mathrm{cpt}}$;

• ${\mathrm{Top}}_{\mathrm{lcpt}}$ for the category of Hausdorff and locally compact topological spaces with morphisms being the proper maps of topological spaces.

The duality itself is exhibited by the following functors

###### Definition

Write

$C:{\mathrm{Top}}_{\mathrm{cpt}}\to {C}^{*}{\mathrm{Alg}}_{\mathrm{com}}^{\mathrm{op}}$C : Top_{cpt} \to C^\ast Alg_{com}^{op}

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

Write

${C}_{0}:*/{\mathrm{Top}}_{\mathrm{cpt}}\to {C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}$C_0 : */Top_{cpt} \to C^\ast Alg_{com,nu}

for the functor that sends $\left(X,{x}_{0}\right)$ to the algebra of continuous functions $f:X\to ℂ$ for which $f\left({x}_{0}\right)=0$.

###### Definition

Write

$\mathrm{sp}:{C}^{*}{\mathrm{Alg}}_{\mathrm{com}}^{\mathrm{op}}\to {\mathrm{Top}}_{\mathrm{cpt}}$sp : C^\ast Alg_{com}^{op} \to Top_{cpt}

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

Similarly write

$\mathrm{sp}:{C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}\to {\mathrm{Top}}_{\mathrm{lcpt}}\phantom{\rule{thinmathspace}{0ex}}.$sp : C^\ast Alg_{com,nu}^{op} \to Top_{lcpt} \,.

## Statement

###### Theorem

(Gelfand duality theorem)

The pairs of functors

${C}^{*}{\mathrm{Alg}}_{\mathrm{com}}^{\mathrm{op}}\stackrel{\stackrel{C}{←}}{\underset{\mathrm{sp}}{\to }}{\mathrm{Top}}_{\mathrm{cpt}}$C^\ast Alg_{com}^{op} \stackrel{\overset{C}{\leftarrow}}{\underset{sp}{\to}} Top_{cpt}

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

###### Corollary

On non-unital ${C}^{*}$-algebras the above induces an equivalence of categories

${C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}\stackrel{\stackrel{{C}_{0}}{←}}{\underset{\mathrm{sp}}{\to }}*/{\mathrm{Top}}_{\mathrm{cpt}}\phantom{\rule{thinmathspace}{0ex}}.$C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} */Top_{cpt} \,.
###### Proof

The operation of unitalization $\left(-{\right)}^{+}$ constitutes an equivalence of categories

${C}^{*}{\mathrm{Alg}}_{\mathrm{nu}}\stackrel{\stackrel{\mathrm{ker}}{←}}{\underset{\left(-{\right)}^{+}}{\to }}{C}^{*}\mathrm{Alg}/ℂ$C^\ast Alg_{nu} \stackrel{\overset{ker}{\leftarrow}}{\underset{(-)^+}{\to}} C^\ast Alg / \mathbb{C}

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

Composed with the equivalence of theorem 1 this yields

${C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}\underset{\simeq }{\overset{\left(-{\right)}^{+}}{\to }}\left({C}^{*}{\mathrm{Alg}}_{\mathrm{com}}/ℂ{\right)}^{\mathrm{op}}\underset{\simeq }{\overset{C}{\to }}*/{\mathrm{Top}}_{\mathrm{cpt}}\phantom{\rule{thinmathspace}{0ex}}.$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

${C}_{0}:*/{\mathrm{Top}}_{\mathrm{cpt}}\underset{\simeq }{\overset{\mathrm{sp}}{\to }}\left({C}^{*}{\mathrm{Alg}}_{\mathrm{com}}/ℂ{\right)}^{\mathrm{op}}\underset{\simeq }{\overset{\mathrm{ker}}{\to }}{C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}$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 $\left(*\stackrel{{x}_{0}}{\to }X\right)$ to $\mathrm{ker}\left(C\left(X\right)\stackrel{{\mathrm{ev}}_{{x}_{0}}}{\to }ℂ\right)$, hence to $\left\{f\in C\left(X\right)\mid f\left({x}_{0}\right)=0\right\}$. This is indeed ${C}_{0}$ from def. 2.

## Generalizations

### In constructive mathematics

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

## References

• 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

An exposition that explicitly gives Gelfand duality as an equivalence of categories and introduces all the notions of category theory necessary for this statement is in

• Ivo Dell’Ambrogio, Categories of ${C}^{*}$-algebras (pdf)

Revised on April 24, 2013 19:54:02 by Urs Schreiber (131.174.42.61)