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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Homework Help: Functional Analysis question

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