I have a commutative Banach algebra A with identity 1. If A contains an element [itex]e[/itex] such that [itex]e^2 = e[/itex] and [itex]e[/itex] is neither 0 nor 1 (I think this also means to say that it contains a non-trivial idempotent), then the maximal ideal space of A is disconnected. Currently I am trying to show this but Im not getting very far. Here is a summary of what I think I may need to show this: Because the question involves the maximal ideal space Im assuming I have to use the Gelfand transform somewhere. In particular it might be interesting to see what the Gelfand transform of the idempotent element e is.