Lemma 3

The mapping $\chi \mapsto$ Ker($\chi$ ) is a bijection of $\mathsf{X}(A)$ onto the set of regular ideals of codimension 1 of A.

Let us recall (A, VIII, p. 426, Definition 1) that an ideal I of A is said to be regular if the quotient algebra $A/I$ admits a unit element.

Let us prove the lemma. On the one hand, $\mathsf{X}(A)$ is identified with the set of characters of $\widetilde{A}$ which are not zero on A. On the other hand, by A, VIII, p. 428, Prop. 4, the mapping $I\mapsto A\cap I$ is a bijection of the set of maximal ideals of $\widetilde{A}$ distinct from A onto the set of regular maximal ideals of A. The lemma then follows from the results of No. 6.

Now suppose that K is a topological field. We then endow $\mathsf{X}'(A)$ with the topology of simple convergence on A; the notation $\mathsf{X}'(A)$ will henceforth denote the topological space thus obtained. When $K =\mathbf{R}$ or $\mathbf{C}$, we shall also call it the weak topology. For every $x\in A$, the function $\mathscr{G}'_A(x)$ on $\mathsf{X}'(A)$ is continuous.

If $h$ is a morphism from A into B, the mapping $\mathsf{X}'(h) :\mathsf{X}'(B)\rightarrow$ $\mathsf{X}'(A)$ is continuous. If $h$ is surjective, $\mathsf{X}'(h)$ is a homeomorphism of $\mathsf{X}'(B)$ onto its image, and this image is closed in $\mathsf{X}'(A)$.

Let $A = A_1\times \cdots \times A_n$; with the same notation as above, $\mathsf{X}'(\pi_i)$ is a homeomorphism of $\mathsf{X}'(A_i)$ onto $\mathsf{X}'_i$ and $\mathsf{X}'_i$ is closed in $\mathsf{X}'(A)$. Hence $\mathsf{X}'_i-\{0\}$ is open in $\mathsf{X}'(A)$; the $\mathsf{X}'(\pi_i)$ define a continuous mapping of the sum space S of the $\mathsf{X}'(A_i)$ onto $\mathsf{X}'(A)$, and one immediately verifies that a union of neighbourhoods of the points $0\in \mathsf{X}'(A_1), . . . ,0\in \mathsf{X}'(A_n)$ has for image a neighbourhood of $0\in \mathsf{X}'(A)$; from all this it follows that $\mathsf{X}'(A)$ is canonically identified with a quotient space of S. In particular, the space $\mathsf{X}(A)$ is identified with the sum space of the $\mathsf{X}(A_i)$.

The canonical bijection of $\mathsf{X}'(A)$ onto $\mathsf{X}(\widetilde{A})$ is a homeomorphism. Let B be a unital algebra over K and $B'$ the underlying algebra; then the space $\mathsf{X}(B)$ is identified with the subspace $\mathsf{X}(B')$ of $\mathsf{X}'(B')$.