Approximate identity

Summary

An approximate identity in a Banach algebra is a net of elements that behaves more and more like a unit. Multiplying by it changes each element less and less. Many important algebras have no unit, such as the convolution algebra $L^1(\R)$ and the algebra of continuous functions vanishing at infinity. In these algebras an approximate identity often replaces the unit in proofs. Every C*-algebra has one, and a bounded approximate identity is the hypothesis of the Cohen factorization theorem.