10. Commutative Banach algebras - Aarhus Universitet

The Gelfand-Mazur theorem
Let A and B be Banach algebras. An isometric isomorphism or just
an isomorphism between A and B is a linear bijection α : A → B
such that α(ab) = α(a)α(b) and α(a) = a for all a,b ∈ A.
We write A ≃ C in this case.
Theorem
(Theorem 10.14) Let A be a unital Banach algebra with the
property that every non-zero element is invertible. Then A ≃ C.
Proof.
Define α : C → A by α(λ) = λ1. Note that α is an algebra
homomorphism and α(λ) = λ1 = |λ|1 = |λ|. It remains
only to show that α is surjective. To this end consider an element
a ∈ A. By Theorem 10.13 there is an element λ in the spectrum of
a. Then λ1 − a is not invertible. Under the present assumption this
implies that a = λ1.
Klaus Thomsen 10. Commutative Banach algebras