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Gelfand duality

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 $\mathrm{sp}(A)$.

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 $C(X)=\{f:X\to ℂ\mid f\mathrm{continuous}\}$, 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 $(X,x_0)$ to the algebra of continuous functions $f:X\to ℂ$ for which $f(x_0)=0$.

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

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

Statement

Theorem

(Gelfand duality theorem)

The pairs of functors

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

is an equivalences of categories.

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

$$C^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}\stackrel{\stackrel{C_0}{\leftarrow }}{\underset{\mathrm{sp}}{\to }}*/{\mathrm{Top}}_{\mathrm{cpt}}.$$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

$$C^{*}{\mathrm{Alg}}_{\mathrm{nu}}\stackrel{\stackrel{\ker }{\leftarrow }}{\underset{(-{)}^{+}}{\to }}{C}^{*}\mathrm{Alg}/ℂ$$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

$$C^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}\underset{\simeq }{\overset{(-{)}^{+}}{\to }}(C^{*}{\mathrm{Alg}}_{\mathrm{com}}/ℂ{)}^{\mathrm{op}}\underset{\simeq }{\overset{C}{\to }}*/{\mathrm{Top}}_{\mathrm{cpt}}.$$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

$$C_0:*/{\mathrm{Top}}_{\mathrm{cpt}}\underset{\simeq }{\overset{\mathrm{sp}}{\to }}(C^{*}{\mathrm{Alg}}_{\mathrm{com}}/ℂ{)}^{\mathrm{op}}\underset{\simeq }{\overset{\ker }{\to }}{C}^{*}{\mathrm{Alg}}_{\mathrm{com},\mathrm{nu}}^{\mathrm{op}}$$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 $(*\stackrel{x_0}{\to }X)$ to $\ker (C(X)\stackrel{{\mathrm{ev}}_{x_0}}{\to }ℂ)$, hence to $\{f\in C(X)\mid f(x_0)=0\}$. 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.

Related concepts

• Isbell duality

  • Gelfand duality

    • C-star algebra, topological space
    • Gelfand spectrum
    • constructive Gelfand duality theorem
    • Serre-Swan theorem

• noncommutative topology

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)

category: analysis, geometry, noncommutative geometry