Banach algebras

Unless we say otherwise, every vector space we talk about is taken to be over C. A Banach algebra is a Banach space A that is also an algebra satisfying ‖AB‖ ≤ ‖A‖ ‖B‖ for A,B ∈ A. We say that A is unital if there is a nonzero element I ∈ A such that AI = A and IA = A for all A ∈ A, called a identity element. If X is a Banach space, let B(X) denote the set of bounded linear operators X → X, and let B0(X) denote the set of compact linear operators X → X (to be compact means that the image of each bounded set is precompact). B(X) is a unital Banach algebra. B0(X) is a Banach algebra, but it is unital if and only if X is finite dimensional. An ideal of a Banach algebra A is a vector subspace I such that