Gelfand transform

Let $A$ be a commutative unital complex algebra and let $\mathrm{\Delta }(A)$ be its character space with the pointwise-evaluation topology (Character and maximal ideal space). For $a\in A$ define its Gelfand transform $\widehat{a}:\mathrm{\Delta }(A)\to \mathbb{C}$ by

$\widehat{a}(\chi ):=\chi (a)(\chi \in \mathrm{\Delta }(A)),$

and define the Gelfand transform of $A$ as the map

${\mathrm{\Gamma }}_A:A⟶{\mathbb{C}}^{\mathrm{\Delta }(A)},{\mathrm{\Gamma }}_A(a):=\widehat{a}\mathrm{.}$

Each $\widehat{a}$ is continuous for the pointwise-evaluation topology, because that topology is by definition the coarsest one making every evaluation $\chi \mapsto \chi (b)$ continuous and $\widehat{a}$ is the evaluation at $a$; the codomain is written ${\mathbb{C}}^{\mathrm{\Delta }(A)}$ with the product topology, and the image of ${\mathrm{\Gamma }}_A$ therefore lies in the algebra $C(\mathrm{\Delta }(A))$ of continuous functions. No supremum norm or compactness is asserted for a general $A$. If $A$ is in addition a nonzero unital commutative Banach algebra and the Axiom of Choice is assumed (The Axiom of Choice), then Maximal ideal space is compact Hausdorff makes $\mathrm{\Delta }(A)$ compact and $C(\mathrm{\Delta }(A))$ carries its supremum norm.

The formula also makes sense if $\mathrm{\Delta }(A)$ is empty: every transform is the unique function on the empty set. It sends $0$ to the zero function. For any character choose $b$ with $\chi (b)\mathrm{\ne }0$; the equality $\chi (b)=\chi (1)\chi (b)$ gives $\chi (1)=1$. Thus the unit maps to the constant-one function (also well defined on an empty character space).

Two qualifications are part of the definition:

• the notation is introduced for unital commutative algebras here; the nonunital version with target $C_0(\mathrm{\Delta }(A))$ for commutative C*-algebras under AC is a theorem proved later (Nonunital commutative Gelfand Naimark) and is not smuggled into the definition;
• no injectivity, surjectivity, isometry or *-preservation is claimed at this point. Those properties are theorems, valid under progressively stronger hypotheses, and the map ${\mathrm{\Gamma }}_A$ is defined for every commutative unital complex algebra.

• Notation. We write $\widehat{a}$ for ${\mathrm{\Gamma }}_A(a)$ and drop the subscript $\mathrm{\Gamma }={\mathrm{\Gamma }}_A$ when the algebra is clear; the algebra, not the element, is what $\mathrm{\Gamma }$ encodes.
• Values are character values. If $A$ is a nonzero commutative unital complex Banach algebra and AC is assumed, the identity ${\sigma }_A(a)=\{\chi (a):\chi \in \mathrm{\Delta }(A)\}$ of Spectrum as character values reads $\mathrm{ran}(\widehat{a})={\sigma }_A(a)$, which is the form in which the spectrum will be computed from the transform.