Proposition 13

Let A be a commutative unital Banach algebra. Let $U_1$ and $U_2$ be open sets of $\mathsf{X}(A)$ forming a partition of $\mathsf{X}(A)$. Then there exists a unique idempotent $j$ of A such that the Gelfand transform $\mathscr{G}(j)$ is equal to 1 on $U_1$ and to 0 on $U_2$.

Let us identify the space $\mathsf{X}(A)$ with a compact subset of $\mathbf{C}^A$ by the mapping $\chi \mapsto (\chi (a))_{a\in A}($cf. No.$^o6$ of I, p. 6 and corollary of Theorem 1 of I, p. 29). The subsets $U_1$ and $U_2$ of the uniform space $\mathbf{C}^A$ are compact and disjoint. By TG, II, p. 31, Proposition 4, there exists a finite subset M of A and disjoint open subsets $V_1$ and $V_2$ of $\mathbf{C}^M$ such that

where $p$ is the canonical projection of $\mathbf{C}^A$ onto $\mathbf{C}^M$.

Let $a_1, . . . , a_n$ be the distinct elements of M, and identify $\mathbf{C}^M$ with $\mathbf{C}^n$. We have Sp$^n_A(a_1, . . . , a_n)\subset p(\mathsf{X}(A))\subset V_1\cup V_2$ since $U_1\cup U_2=\mathsf{X}(A)$. Let $f$ be the function on $V_1\cup V_2$ equal to 1 on $V_1$ and to 0 on $V_2$. We have $f\in \mathscr{O}(V_1\cup V_2)$. Put $j=f(a_1, . . . , a_n)$. Since $f^2=f$, we have $j^2=j$. By corollary 1 of I, p. 66, we have $\chi (j) = 1$ if $\chi \in U_1$ and $\chi (j) = 0$ if $\chi \in U_2$, which proves the existence of the required idempotent.

Moreover, if $j_1$ is an idempotent of A, the relations $j^2=j$ and $j_1^2=j_1$ imply $(j-j_1)(j+j_1-1) = 0$. If $\mathscr{G}(j_1) =\mathscr{G}(j)$, the Gelfand transform of $j+j_1-1$ has its values in $\{-1,1\}$, hence $j+j_1-1$ is invertible (Proposition 6 of I, p. 37), whence $j=j_1$.