---
tag: NT564Y315M
kind: statement
role: theorem
name: Commutative Gelfand–Naimark theorem
msc: [46L05, 46J05]
uses: [M7JVFJ3MMX, VWNSHKJT92, N59BW1QFD1, T4RFYT8HDN, 1P8D3MK79H]
about: [VWNSHKJT92]
sources:
  - url: https://en.wikipedia.org/wiki/Gelfand_representation
  - url: https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark_theorem
---

## Statement

Let $A$ be a commutative C*-algebra, and let $\Delta(A)$ be its [[character space|M7JVFJ3MMX]].
Let $C_0(\Delta(A))$ be the C\*-algebra of [[continuous functions vanishing at infinity|N59BW1QFD1]] on $\Delta(A)$, with the sup norm and complex conjugation.
Then the [[Gelfand transform|VWNSHKJT92]] $\Gamma \colon A \to C_0(\Delta(A))$, $\Gamma(a)(\varphi) = \varphi(a)$, is an isometric $*$-isomorphism of $A$ onto $C_0(\Delta(A))$.

## Remarks

If $A$ has a unit, $\Delta(A)$ is compact and $A \cong C(\Delta(A))$.
Proof idea: a character takes real values on self-adjoint elements, so $\Gamma(a^*) = \overline{\Gamma(a)}$.
For self-adjoint $a$ the C\* identity gives $\norm{a^{2^k}} = \norm{a}^{2^k}$, so by [[Gelfand's spectral radius formula|T4RFYT8HDN]] $\norm{\Gamma(a)}_\infty = \norm{a}$; applying this to $a^* a$ shows that $\Gamma$ is isometric for all $a$.
So the range of $\Gamma$ is a closed $*$-subalgebra that separates points and vanishes at no point, and by the [[Stone–Weierstrass theorem|1P8D3MK79H]] it is all of $C_0(\Delta(A))$.

Example: for $A = C(X)$ with $X$ compact Hausdorff, $\Delta(A)$ is homeomorphic to $X$ through the evaluations, and $\Gamma$ is the identity map up to this homeomorphism.

In the unital case, the theorem gives a contravariant equivalence between commutative unital C\*-algebras with unital $*$-homomorphisms and compact Hausdorff spaces with continuous maps.

## Explanations

### L1 human

Every commutative C\*-algebra is an algebra of continuous functions on some space, with nothing lost.
The space is recovered from the algebra as the set of its characters.
So studying commutative C\*-algebras is the same as studying locally compact Hausdorff spaces.

### L2 human

This theorem is the starting point of "noncommutative topology": a general C\*-algebra is thought of as the functions on a space that does not exist.
It gives the [[continuous functional calculus|JB7S6WPZ77]].
For a normal element $a$, the C\*-algebra generated by $a$ and $1$ is commutative, and its character space is the spectrum of $a$.
So it is isomorphic to $C(\sigma(a))$.
The noncommutative Gelfand–Naimark theorem says that every C\*-algebra is isometrically $*$-isomorphic to a closed $*$-subalgebra of the bounded operators on a Hilbert space.