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Maximal Ideals and Maximal Subspaces in normed algebras

  1. Jul 20, 2014 #1
    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.
     
  2. jcsd
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