Banach Algebras

The aim of this notes is to provide basic information about commutative Banach algebras. The final goal is to show that a unital, commutative complex Banach algebra A can be embedded as subalgebra of C(MA), the algebra of continuous functions on a w∗-compact setMA, known as the maximal ideal space or character space. Also, the non unital commutative complex Banach algebra can be embedded as a subalgebra of C0(Ω), the algebra of continuous functions on a w∗-locally compact (but not compact) set Ω, vanishing at ∞. Definition 1.1 (Algebra). Let A be a non-empty set. Then A is called an algebra if (1) (A,+, .) is a vector space over a field F (2) (A,+, ◦) is a ring and (3) (αa) ◦ b = α(a ◦ b) = a ◦ (αb) for every α ∈ F, for every a, b ∈ A Usually we write ab instead of a ◦ b for notational convenience. Definition 1.2. An algebra A is said to be (1) real or complex according to the field F = R or F = C respectively. (2) commutative if (A,+, ◦) is commutative