A Banach algebra generated by idempotents i.e. elements x such that x^{2} = x.

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In the following, we show that the Banach algebra C(X), where X is a compact Hausdorff space, with Card(X) > 1 ,is generated by idempotents iff X is totally disconnected. Recall that a topological space is said to be totally disconnected if for every distinct x_{1}, x_{2} ∈ X, there exist disjoint open sets G_{1} and G_{2} such that x_{1} ∈ G_{1}, x_{2} ∈ G_{2} and X = G_{1} ∪ G_{2}. If X is totally disconnected, x_{1} ≠ x_{2}, x_{1} ∈ G_{1}, x_{2} ∈ G_{2}, X = G_{1} ∪ G_{2}, G_{1} ∩ G_{2} = ∅, G_{1} and G_{2} are open, then the continuous function f(x) = \{ \begin{matrix}1 & x ∈ G_{1} \\ 0 & x ∈ G_{2} \end{matrix} separates x_{1} and x_{2}. So the closed self-adjoint subalgebra generated by idempotent, by the Stone-Weierstrass theorem, is C(X). Conversely, suppose that C(X) is generated by its idempotents. Let x_{1} and x_{2} belong to X. By Urysohn’s lemma there exists a function f ∈ C(X) such that f(x_{1}) = 1 and f(x_{2}) = 0. Every element of the self-adjoint subalgebra generated by idempotents is of the form h = ∑_{i=1}^{k}λ_{i}g_{i}(♣) for some idempotents g_{i} and λ_{i} ∈ C. Hence there is a sequence (h_{n}) of elements of the form (♣) such that h_{n} ⟶ f uniformly on X. So h_{n}(x_{1}) ⟶ 1 and h_{n}(x_{2}) ⟶ 0. Therefore there exists a number N such that |h_{N}(x_{1})| > \frac{1}{2} and |h_{N}(x_{2})| < \frac{1}{2}. So that x_{1} ∈ h_{N}^{−1}({z ∈ C; |z| > 1}) = G_{1}, x_{2} ∈ h_{N}^{−1}({z ∈ C; |z| < 1}) = G_{2}, X = G_{1} ∪ G_{2}, X = G_{1} ∩ G_{2} = ∅. Thus X is totally disconnected.