Thread: 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?

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 is the inverse of . In other words, . The comment about the spectrum then follows immediately from the fact that is invertible.

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.