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Math Help - Banach algebras question 4

  1. #1
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    Banach algebras question 4

    I've the lats three questions about maximal ideals in Banach algebras.

    How can I show the following (in red shape) part of proof?




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  2. #2
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    Quote Originally Posted by Arczi1984 View Post
    I've the lats three questions about maximal ideals in Banach algebras.

    How can I show the following (in red shape) part of proof?



    I'm not sure what the difficulty is here. The first sentence of the proof shows that if x is invertible then T(x^{-1}) is the inverse of Tx. In other words, (Tx)^{-1} = T(x^{-1}). The comment about the spectrum then follows immediately from the fact that \lambda\notin\sigma(x)\;\Leftrightarrow\;x-\lambda e is invertible.
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  3. #3
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    Here is my mistake. When I read it I saw non-invertible I spent to much time over this paper and now I see problems where everything is clear. I'm sorry for this question and thanks for answer.
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