In the words of Guillemin and Sternberg, the key idea in Gelfand theory can be summarized as

Maximal ideals are the underlying ‘points’ of a commutative normed ring.

Gelfand theory can be seen as the generalization of Fourier theory, in that for commutative group algebras the Gelfand transform coincides with the Fourier transform.

## Characters

Similar to the character of a group, the character for an algebra $A$ is, roughly speaking, an approximation of representations of $A$, and the character space (also called *spectrum*) for $A$ the archetype of the unitary dual of a $C^\ast$-algebra.

*character*on $A$ is a non-zero homomorphism $\phi:A\to \mathbb{C}$.

These are also called *multiplicative linear functionals*. The set $\Omega(A)$ of all characters on $A$ and call it *character space* of $A$. An example for a character is $\operatorname{ev}_{x_0}\in \Phi_{C(\Omega)}$ where $\operatorname{ev}_{x_0} f = f(x_0)$ for $f\in A$.

*Proof:*Clearly for each $\phi\in\Omega(A)$, $\operatorname{Ker}(\phi)$ is a maximal ideal of codimension $1$. The map is injective, since for $\operatorname{Ker}(\phi) = \operatorname{Ker}(\psi)$, $\phi(a-\psi(a)) = 0$ (notice that $a-\psi(a) \in \operatorname{Ker}(\psi)$ by linearity) for all $a\in A$ and hence $\phi = \psi$. The map is surjective, since given a maximal ideal $M$ of codimension 1, the quotient $\phi: A\to A/M = \mathbb{C}$ is a character with $\operatorname{Ker}(\phi) = M$. □

For unital Banach algebras, the character space $\Omega(A)$ can be endowed with a weak $\ast$-topology, *Gelfand topology*, and be made into a compact topological space, also called the character space, or the *spectrum* of the algebra $A$, which is also denoted by $\Omega(A)$. This is possible by the following theorem

*Proof:*$\phi(a)\in \sigma(a)$ where $\sigma(a)$ is the spectrum of $a\in A$. This means $|\phi(a)|\leq r(a) \leq \|a\|$ where $r(a)$ is the spectral radius, so $\|\phi\|\leq 1$ since $\|a\| \leq 1$ (recall that $||a|| = \lim_{n\to \infty}\|a^n\|^{1/n}$). Now $\phi(1) = \phi(1)^2$ means $\phi(1)=1$, so $\|\phi\| = 1$. □

*state space*of $A$ ($A’$ is the space of linear functions on $A$, not the commutant). Endow the space $A’$ with the weak $\ast$-topology (the topology defined by $\lambda_\nu \to \lambda$ iff $\lambda_\nu(a) \to \lambda(a)$ for each $a\in A$), the

*Gelfand topology*on $\Omega(A)$ is the relative weak $\ast$-topology from $A’$. $\Omega(A)$ is compact in this topology.

## Gelfand transform/representation

Now for genereal unital Banach algebra $A$ the spectrum $\Omega(A)$ might be empty, but for *commutative* Banach algebras the *Gelfand representation theorem* holds, of which a corollary is that any semisimple commutative unital Banach algebra is a subalgebra of $C(\Omega(A)$.

For $a$ in $A$ a commutative Banach algebra, let $\hat{a}$ denote the *Gelfand transform* of $a$, defined on $\Omega(A)$ by
$$
\hat{a}:\Omega(A) \to \mathbb{C}: \phi\mapsto\phi(a).
$$
The topology on $\Omega(A)$ is the smallest one making all the $\hat{a}$ continuous. In fact $\hat{a}\in C_0(\Omega(A))$.

- The map $A\to C_0(\Omega(A)): a\to \hat{a}$ is a norm-decreasing homomorphism,
- $\sigma(a) = \hat{a}(\Omega(A))$,
- $r(a) = \|\hat{a}\|_\infty$ where $\|\cdot\|$ is the sup norm,
- $a$ is invertible iff $\hat{a}$ is invertible
- $\operatorname{rad}(A) = \mathcal{Q}(A) = \operatorname{Ker}(A\to C_0(\Omega(A))$ where $\mathcal{Q}(A)$ is the set of quasinilpotent elements (those with spectrum $\\{0\\}$).

- if $A$ is semisimple, $\operatorname{Ker}(A\to C_(\Omega(A)))$ is trivial, $A\to C_(\Omega(A))$ is injective, making $A$ into a subalgebra of $C_0(\Omega(A))$.
- $\hat{a}$ should really be seen as $\hat{a}: \Omega(A) \to \sigma(a)$.

## $C^\ast$-algebras

The Gelfand transform embeds $A$ in $C_0(\Omega(A))$ for $A$ a commutative unital semisimple Banach algebra. This is a surjection for $C^\ast$-algebras.

This is a special case for the general Gelfand-Naimark theorem, which says that every $C^\ast$-algebra is isometrically $\ast$-isomorphic to a $C^\ast$-subalgebra of bounded operators on a Hilbert space.