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Math Help - Banach algebras question 2

  1. #1
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    Banach algebras question 2

    Next questions about proof details.
    Red shape 1: From where we have this?
    Red shape 2: How we used fact that this algebra is semi-simple?



    PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
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  2. #2
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    1. The fact that  f(Tx)\in\sigma(Tx) is just condition (2) from earlier in the paper (which holds because f is multiplicative).

    2. Equation (10) says that f(Txy) = f(TxTy) for each multiplicative linear functional f on the algebra. The semisimplicity condition says that there are enough such functions to separate elements of the algebra. So if T(xy) and T(x)T(y) have the same value at each f then they must be equal.
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