approximate identity

Let $𝒜$ be a Banach algebra.

A left approximate identity for $𝒜$ is a net ${(e_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ in $𝒜$ which :

1. , for some constant $C$.

2. $e_{\lambda }a⟶a$, for every $a\in 𝒜$.

Similarly, a right approximate identity for $𝒜$ is a net ${(e_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ in $𝒜$ which :

1. , for some constant $C$.

2. $a{e}_{\lambda }⟶a$, for every $a\in 𝒜$.

An approximate identity for a $𝒜$ is a net ${(e_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ in $𝒜$ which is both a left and right approximate identity.

0.0.1 Remarks:

• There are examples of Banach algebras that do not have approximate .

• If $𝒜$ has an identity element $e$, then clearly $e$ itself is an approximate identity for $𝒜$.