# nLab Gelfand duality

### 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 (locally) compact topological spaces and commutative (nonunital) 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 C-star algebras;

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

• $C^\ast Alg_{com} \subset C^\ast Alg$ for the full subcategory of commutative $C^\ast$-algebras;

• $C^\ast Alg_{com,nu} \subset C^\ast Alg_{nu}$ for the full subcategory of commutative non-unital $C^\ast$-algebras.

And

The duality itself is exhibited by the following functors

###### Definition

Write

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

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

Write

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

for the functor that sends $(X,x_0)$ to the algebra of continuous functions $f : X \to \mathbb{C}$ 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

(Gelfand duality theorem)

The pairs 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 1 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. 2.

## 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.

## 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 $C^\ast$-algebras (pdf)

Some other generalized contexts for Gelfand duality:

• Hans-E. Porst, Manfred B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta mathematica 25:2, (1978) pp 169-204
• H. Heunen, N. P. Landsman, Bas Spitters, S. 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

Revised on April 9, 2014 05:57:20 by Tim Porter (2.26.27.237)