Definition 5

Let A be a commutative unital algebra over K. For every $x\in A$, let $\mathscr{G}_A(x)$, or simply $\mathscr{G}(x)$, denote the mapping $\chi \mapsto$ $\chi (x)$ of $\mathsf{X}(A)$ into K. It is called the Gelfand transform of $x$.

The mapping $\mathscr{G}$ is a unital morphism of A into the unital algebra $K^{\mathsf{X}(A)}$ of mappings from $\mathsf{X}(A)$ into K. It is called the Gelfand transform of A.

If $x\in$ A, the image of the Gelfand transform $\mathscr{G}_A(x)$ of $x$ is contained in Sp$_A(x)$. Indeed, let $\chi \in \mathsf{X}(A)$; since $\chi (x-\chi (x)e) =$ 0, the element $x-\chi (x)e$ is not invertible.

Let B be a commutative unital algebra over K and $h$ a unital morphism of A into B; then $\mathsf{X}(h) :\mathsf{X}(B)\rightarrow \mathsf{X}(A)$ defines a unital morphism $h_*: K^{\mathsf{X}(A)}\rightarrow K^{\mathsf{X}(B)}$, and the diagram:

$\mathscr{G}\leftarrow_A\mathsf{X}(A)$

A $\rightarrow K$ (3) $\rightarrow \leftarrow_h\rightarrow \leftarrow_{h_*}$

$\mathscr{G}\leftarrow_B\mathsf{X}(B)$

B $\rightarrow K$

is commutative. Indeed, for every $x\in A$ and every $\chi \in \mathsf{X}(B)$, we have:

Now suppose that K is a topological field. One then endows $\mathsf{X}(A)$ with the topology of simple convergence on A (cf. EVT, III, p. 14, Example 1), and the topological space $\mathsf{X}(A)$ is called the space of characters of A. The topology of $\mathsf{X}(A)$ is therefore the least fine for which the functions $\mathscr{G}_A(x)$ for $x\in A$ are continuous, and the mapping $\chi \mapsto (\chi (a))_{a\in A}$ identifies the space $\mathsf{X}(A)$ with a subset of $K^A$.

When K = $\mathbf{R}$ or $\mathbf{C}$, this topology is the topology induced on $\mathsf{X}(A)\subset A^*$ by the weak topology $\sigma (A^*,A)$ on $A^*$ (EVT, II, p. 45, Def. 2); in this respect, we shall also say that it is the weak topology on $\mathsf{X}(A)$.

If $h$ is a unital morphism of A into B, the mapping $\mathsf{X}(h) :\mathsf{X}(B)\rightarrow \mathsf{X}(A)$ is continuous. If $h$ is surjective, the image of $\mathsf{X}(h)$, namely the set of characters of A vanishing on the kernel of $h$, is closed in $\mathsf{X}(A)$; on the other hand, the topology on $\mathsf{X}(h)(\mathsf{X}(B))$ deduced from that of $\mathsf{X}(B)$ by the bijection $\mathsf{X}(h)$ is the topology of simple convergence in A, that is to say the topology induced by that of $\mathsf{X}(A)$; in other words, $\mathsf{X}(h)$ is a homeomorphism of $\mathsf{X}(B)$ onto a closed subset of $\mathsf{X}(A)$.

If $A_1, . . . ,A_n$ are commutative unital algebras over K, the space $\mathsf{X}(A_1\times \cdots \times A_n)$ is thus identified with the topological sum of $\mathsf{X}(A_1), . . . ,\mathsf{X}(A_n)$. Analogously, $\mathsf{X}(A_1\otimes \cdots \otimes A_n)$ is identified with the topological product $\mathsf{X}(A_1)\times \cdots \times \mathsf{X}(A_n)$.