(a) A unital Banach algebra, except (a) A unital Banach algebra, except the algebra of complex numbers, without nontrivial idempotent.
(b) A unital Banach algebra with a nontrivial idempotent.
(Recall that 0 and 1 are called trivial idempotents.)

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We show that the Banach algebra C(X) has no nontrivial idempotent iff X is connected:
Let 0 ¹ f ¹ 1 be an idempotent. Then X = f-1({0}) Èf-1({1}) implies that X is not connected. Conversely if X is disconnected and X = G1 ÈG2 with open disjoint sets G1 and G2, then f(x) = {
1
x Î G1
0
x Î G2
is a trivial idempotent of C(X).

Comment. If A is a (not necessarily commutative) Banach algebra with an element a Î A such that sp(a) is not connected, then A has a nontrivial idempotent. (cf. [B&D, Remarks of Prop. 7.9 ])

Ref.

[B&D] F.F. Bonsall, J. Duncan, complet normed algebras, Springer-Verlag,1973.

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On 22 Feb 2001, 00:13.