# nLab Gelfand spectrum

## Theorems

### Euclidean QFT

#### Topology

topology

algebraic topology

# Contents

## Idea

The Gelfand spectrum (originally Гельфанд) of a commutative C*-algebra $A$ is a topological space $X$ such that $A$ is the algebra of complex-valued continuous functions on $X$. (This “Gelfand duality” is a special case of the general duality between spaces and their algebras of functions.)

## Definition

###### Definition

Given a unital, not necessarily commutative, complex C*-algebra $A$, the set of its characters, that is: continuous nonzero linear homomorphisms into the field of complex numbers, is canonically equipped with what is called the spectral topology which is compact Hausdorff. (If applied to a nonunital $C^*$-algebra, then it is only locally compact.) This correspondence extends to a functor, called the Gel’fand spectrum from the category C*Alg of unital $C^*$-algebras to the category of Hausdorff topological spaces.

###### Remark

A character on a unital Banach algebra is automatically a continuous function (with Lipschitz constant 1).

###### Remark

The Gelfand spectrum functor is a full and faithful functor when restricted to the subcategory of commutative unital $C^*$-algebras.

###### Remark

The kernel of a character is clearly a codimension-$1$ closed subspace, and in particular a closed maximal ideal in $A$; therefore the Gel’fand spectrum is a topologised analogue of the maximal spectrum of a discrete algebra.

###### Remark

For noncommutative $C^*$-algebras the spaces of equivalence classes of irreducible representations (i.e., the spectrum) and their kernels (i.e., the primitive ideal space) are more important than the character space.

###### Remark

The Gelfand spectrum is also useful in the context of more general commutative Banach algebras.

category: analysis

Revised on November 21, 2013 11:37:59 by Urs Schreiber (188.200.54.65)