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 $\displaystyle T(x^{-1})$ is the inverse of $\displaystyle Tx$. In other words, $\displaystyle (Tx)^{-1} = T(x^{-1})$. The comment about the spectrum then follows immediately from the fact that $\displaystyle \lambda\notin\sigma(x)\;\Leftrightarrow\;x-\lambda e$ is invertible.