Let $\mathcal{A}$ be a Banach algebra.

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

1. $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$ , for some constant $C$ .
2. $e_{\lambda}a \longrightarrow a\;$ , for every $a \in \mathcal{A}$ .

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

1. $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$ , for some constant $C$ .
2. $ae_{\lambda} \longrightarrow a\;$ , for every $a \in \mathcal{A}$ .

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

Remarks:

• There are examples of Banach algebras that do not have approximate identities.
• If $\mathcal{A}$ has an identity element $e$ , then clearly $e$ itself is an approximate identity for $\mathcal{A}$ .