# Homework Help: Maximal Ideals and Maximal Subspaces in normed algebras

1. Jul 20, 2014

### Murtuza Tipu

1. The problem statement, all variables and given/known data
Let A be a unital commutative Banach Algebra and I a maximal ideal of A. Prove that I is a maximal subspace of A. Is this result still valid if A is not Banach or commutative or unital?

2. Relevant equations

The first part is pretty easy: Maximal ideals are of the form kerτ for some character τ:A→C, and then A=kerτ+C1, so kerτ is a maximal subspace.

If A is not commutative, we have a counter-example: The algebra M2(C) is simple, so the unique maximal ideal of M2(C) is {0}, which is not a maximal subspace.

3. The attempt at a solution
I couldn't solve the rest of the question: If A is Banach and commutative but non-unital, maybe I could consider its unitization and associate maximal ideals of A and A˜, but I couldn't do that.

The only example of commutative, unital, non-Banach normed algebra I can think of is C[x], the polynomial algebra in one variable (with any of the usual norms), but the only maximal ideals of C[x] are of the form p(x)C[x] with degree(p)=1, and these are maximal subspaces.

Any hints are appreciated. Thank you.