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Math Help - Maximal ideals in Banach algebras

  1. #1
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    Maximal ideals in Banach algebras

    Hi!
    I've a little problem with understanding proof of the following theorem:

    In the red frames are this steps with which I've problems. In the first one I don't know from where we know that X is a zero set. In the second frame - how can I explain this in the simplest way.
    Maybe there are simple questions but I'm rather algebraist and I always have problems with functionala analysis
    I'll be grateful for any help.
    Best regards
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  2. #2
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    Opalg's Avatar
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    For the first question, if X has codimension 1, it means that you can fix an element z\notin X and then any element y in A can be uniquely expressed in the form y = x + \lambda z, where x\in X and \lambda is a scalar. Define f(y) = \lambda. Then f is a linear functional whose zero set is X.

    For the second question, if y=x-f(x)e then f(y) = f(x) -f(x)=0 (since f(e)=1). Thus y is in the zero set of f, which by hypothesis consists of non-invertible elements. But if an element is non-invertible then 0 is in its spectrum.
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