# Spectrum space of semidirect product of a subalgebra and an ideal of a Banach Algebra

If an algebra $A$ is a semidirect product of a subalgebra $B$ and an ideal $I$. Is characterized the character space of $A$ by character space of $B$ and character space of $I$?

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Am I correct in guessing that words like "character amenability" and "module extension Banach algebra" are lurking in the background to your question? –  Yemon Choi Mar 12 '13 at 21:31
My guess is that the solution should not be too difficult and is the sort of thing that people in the past were happy to rediscover as and when they needed it. Why don't you tell us how far you have got at the moment? –  Yemon Choi Mar 12 '13 at 21:34
I think that the spectrum space of the semidirect product of a subalgebra B and an closed two sided ideal I of a Banach algebra A is \$\sigma(B)union{0}×\sigma(I)union {0}. I proved one direction and I haven't proved the other direction –  Ali Mar 12 '13 at 22:15
Assuming you are a student, what does your supervisor say? –  Yemon Choi Mar 14 '13 at 19:00
Ali, why on earth not? That is what a supervisor is for –  Yemon Choi Mar 14 '13 at 23:31