---
tag: 81ZC98VMD5
kind: definition
name: Gelfand transform
msc: [46J05, 46J25]
defines: VWNSHKJT92
uses: [M7JVFJ3MMX, N59BW1QFD1]
sources:
  - url: https://en.wikipedia.org/wiki/Gelfand_representation
---

## Definition

Let $A$ be a commutative complex Banach algebra, and let $\Delta(A)$ be its [[character space|M7JVFJ3MMX]].
For $a \in A$, the *Gelfand transform* of $a$ is the function
$$\hat a \colon \Delta(A) \to \C, \qquad \hat a(\varphi) = \varphi(a).$$
The *Gelfand transform* (or Gelfand representation) of $A$ is the map $\Gamma \colon A \to C_0(\Delta(A))$, $a \mapsto \hat a$, into the [[continuous functions vanishing at infinity|N59BW1QFD1]] on $\Delta(A)$.

## Remarks

Each $\hat a$ is continuous by the definition of the weak-* topology, and it vanishes at infinity.
The map $\Gamma$ is an algebra homomorphism with $\norm{\hat a}_\infty \le \norm{a}$.
If $A$ has a unit, then $\hat a(\Delta(A)) = \sigma(a)$, the spectrum of $a$, so $\norm{\hat a}_\infty$ is the spectral radius of $a$.
The kernel of $\Gamma$ is the Jacobson radical of $A$.

Example: let $A = \ell^1(\Z)$ with convolution $(a * b)_n = \sum_k a_k b_{n-k}$.
For $z$ in the unit circle, $\varphi_z(a) = \sum_n a_n z^n$ is linear, nonzero, and multiplicative by the Cauchy product formula, so it is a character.
Every character has this form, and $z \mapsto \varphi_z$ identifies $\Delta(A)$ with the circle.
The Gelfand transform of $a$ is the absolutely convergent Fourier series $\hat a(z) = \sum_n a_n z^n$.

Non-example: the construction needs $A$ to be commutative.
The algebra $M_2(\C)$ has no characters at all.
A character vanishes on commutators, and the commutators span the trace-zero matrices.
So its kernel is a nonzero two-sided ideal of the simple algebra $M_2(\C)$, hence everything.
So for $M_2(\C)$ the "Gelfand transform" would map into functions on the empty set and lose all information.