10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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<strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong> - the definition<br />
Let A be a <strong>Banach</strong> algebra. A is commutative or abelian when<br />
ab = ba for all a,b ∈ A.<br />
Let A be a commutative <strong>Banach</strong> algebra. An ideal in A is a<br />
subspace J ⊆ A such that aj ∈ J when a ∈ A and j ∈ J.<br />
Lemma<br />
(Proposition 11.2) Let A be a commutative and unital <strong>Banach</strong><br />
algebra.<br />
(a) The only ideal in A which contains an invertible element is A<br />
itself.<br />
(b) If J ⊆ A is an ideal then the closure J of J is also an ideal in A.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>