Exercise 33, § 2

Let $A$ be a commutative unital Banach algebra over $\mathbf{C}$. Suppose that every closed ideal I of A is finitely generated (that is, there exist $n\in \mathbf{N}$ and $x_1, . . . , x_n\in A$ such that $I = Ax_1+\cdots + Ax_n$).

a) Every ideal I of A is closed. (Write $I = Ax_1+\cdots +Ax_n$ with $x_1, . . . , x_n$ in A. Let $(x_{ip})_{p\geqslant 1}$ be a sequence of elements of I tending to $x_i$. For every $(y_1, . . . , y_n)\in A^n$ and every $p\geqslant 1$, put

We have $\varphi \in \mathscr{L}(A^n,I),\varphi_p\in \mathscr{L}(A^n,I),\|\varphi -\varphi_p\|$ tends to 0 when $p$ tends to $+\infty$, and $\varphi$ is surjective. Deduce that $\varphi_p$ is surjective for $p$ sufficiently large, by considering the transposes of $\varphi$ and $\varphi_p$. Hence I = I.)

b) If A is integral, $A =\mathbf{C}\cdot 1$. (Use a) and Exerc. 15.)

c) If the only nilpotent element of A is 0, there exists an integer $n\geqslant 0$ such that A is isomorphic to $\mathbf{C}^n$. (Since A is noetherian, $\{0\}$ is intersection of prime ideals $\mathfrak{p}_1, . . . ,\mathfrak{p}_m$; these are closed by a), and the algebras $A/\mathfrak{p}_i$ are isomorphic to $\mathbf{C}$ by b).)

d) One has dim$_{\mathbf{C}}A<+\infty$. (Let N be the set of nilpotent elements of A, which is a (closed) ideal of A. Then dim(A$/N$)$<+\infty$ by c). Since N is a finitely generated ideal, $N^n= 0$ for $n$ sufficiently large. Finally, each $N^i/N^{i+1}$ is a finitely generated module over $A/N$, hence dim(N$^i/N^{i+1}$)$<+\infty$.)