10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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Theorem 11.5<br />
Proof.<br />
a) Let J be a maximal ideal in A. Then J is closed by b) of<br />
Theorem 11.3. Hence J is a closed two-sided ideal in A, and we<br />
conclude that A/J is a <strong>Banach</strong> algebra by Lemma 5.<br />
We assert that every non-zero element of A/J is invertible. To see<br />
this, let z ∈ A/J be a non-zero element. Then<br />
z (A/J) = {zξ : ξ ∈ A/J} is an ideal in A/J, and it follows that<br />
q −1 (z (A/J))<br />
is an ideal in A.<br />
Note that J ⊆ q −1 (z (A/J)) and that J = q −1 (z (A/J)). Since J<br />
is maximal ideal, q −1 (z (A/J)) can then not be a proper ideal in A,<br />
i.e. A = q −1 (z (A/J)). In particular, 1 = q(1) ∈ z (A/J), proving<br />
that z is invertible.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>