Corollary

Let A be a commutative unital Banach algebra. The following assertions are equivalent:

a) The space of characters $\mathsf{X}(A)$ is not connected;

b) There exists an idempotent element of A different from 0 and 1;

c) The algebra A is isomorphic to the product of two non-zero Banach algebras.

The proposition proves that a) implies b). If $j$ is an idempotent of A, let $I_1=jA$ and $I_2= (1-j)A$. Then $I_1$ and $I_2$ are closed ideals of A, and $I_1+ I_2= A$. If $j \notin \{0,1\}$, the ideals $I_1$ and $I_2$ are distinct from A. On the other hand, the ideal $I_1$ (resp. $I_2$) is the set of elements $x$ of A such that $jx=x$ (resp. $(1-j)x=x$), hence $I_1\cap I_2=\{0\}$. The algebra A is therefore identified with the product $A/I_1\times A/I_2$. Thus assertion b) implies c). Finally, if A is isomorphic to $A_1\times A_2$, the space $\mathsf{X}(A)$ is identified with the sum space of $\mathsf{X}(A_1)$ and of $\mathsf{X}(A_2)$ (I, p. 6, n$^o6$), hence c) implies a).