Definition 6

Let A be a commutative algebra over K. A character of A is, by definition, an algebra morphism of A into K.

The set of characters of A will be denoted by $\mathsf{X}'(A)$.

The zero mapping is an algebra morphism. If A has a unit element $e$, a nonzero algebra morphism of A into K is unital, that is to say, is a unital character of K in the sense of Definition 4: indeed, in order that $\chi \in \mathsf{X}'(A)$ be nonzero, it is necessary and sufficient that $\chi (e) = 1$.

We shall put $\mathsf{X}(A) =\mathsf{X}'(A)-\{0\}$; from the foregoing, this notation is compatible with that introduced when A is unital.

If $h: A\rightarrow B$ is a morphism of commutative algebras, the mapping $\chi \mapsto \chi \circ h$ is a mapping $\mathsf{X}'(h) :\mathsf{X}'(B)\rightarrow \mathsf{X}'(A)$. It maps 0 to 0. If $k: B\rightarrow C$ is a morphism of commutative algebras, then one has $\mathsf{X}'(k\circ h) =\mathsf{X}'(h)\circ \mathsf{X}'(k)$. If $h$ is surjective, $\mathsf{X}'(h)$ is a bijection of $\mathsf{X}'(B)$ onto the set of characters of A vanishing on the kernel of $h$. Let $A_1, . . . ,A_n$ be commutative algebras, $A = A_1\times \cdots \times A_n$ and $\pi : A\rightarrow A_i$ the canonical morphism; then $\mathsf{X}'(\pi_i)$ is a bijection of $\mathsf{X}'(A_i)$ onto a subset $\mathsf{X}'_i$ of $\mathsf{X}'(A)$, namely the set of characters of A vanishing on $\prod_{j\not=i}A_j$; one sees as in No.$^o6$ that $\mathsf{X}'(A)$ is the union

of the $\mathsf{X}'_i$; on the other hand, $\mathsf{X}'_i\cap \mathsf{X}'_j=\{0\}$ for $i\not=j$; in particular the $\mathsf{X}'_i-\{0\}$ form a partition of $\mathsf{X}'(A)-\{0\}=\mathsf{X}(A)$.

For every $x\in A$, let $\mathscr{G}'_A(x)$, or simply $\mathscr{G}'(x)$, be the mapping $\chi \mapsto \chi (x)$ of $\mathsf{X}'(A)$ into K. The mapping $\mathscr{G}'$ is a morphism of A into the algebra $A_1$ of mappings $\mathsf{X}'(A)\rightarrow K$ vanishing at 0. Let B be a commutative algebra, $B_1$ the algebra of mappings $\mathsf{X}'(B)\rightarrow$ K vanishing at 0, and $h$ a morphism of A into B; then $\mathsf{X}'(h)$ defines a morphism $h_1: A_1\rightarrow B_1$, and one has $h_1\circ \mathscr{G}'_A=\mathscr{G}'_B\circ h$. One denotes by $\mathscr{G}_A(x)$, or simply $\mathscr{G}(x)$, the restriction of $\mathscr{G}'_A(x)$ to $\mathsf{X}(A)$, and calls it the Gelfand transform of $x$.

Let $\widetilde{A}$ be the unital algebra deduced from A by adjunction of a unit element. By restriction, every character of $\widetilde{A}$ defines a character of A; conversely, every character of A extends in a unique way to a character of $\widetilde{A}$. This defines a canonical bijection of $\mathsf{X}'(A)$ onto $\mathsf{X}(\widetilde{A})$, by which these two sets are identified. The character 0 of A is identified with the unique character of $\widetilde{A}$ with kernel A.

If $x\in A$ and $\chi \in \mathsf{X}'(A)$, one has $\chi (x)\in$ Sp$_{\widetilde{A}}(x)$, hence $\chi (x)\in$ Sp$'_A(x)$.