# Math Help - Banach algebras question 4

1. ## 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?

2. Originally Posted by Arczi1984
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.

3. 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.