approximate identity
Let $\mathcal{A}$ be a Banach algebra^{}.
A left approximate identity for $\mathcal{A}$ is a net ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$ in $\mathcal{A}$ which :

1.
$$, for some constant $C$.

2.
${e}_{\lambda}a\u27f6a$, for every $a\in \mathcal{A}$.
Similarly, a right approximate identity for $\mathcal{A}$ is a net ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$ in $\mathcal{A}$ which :

1.
$$, for some constant $C$.

2.
$a{e}_{\lambda}\u27f6a$, for every $a\in \mathcal{A}$.
An approximate identity for a $\mathcal{A}$ is a net ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$ in $\mathcal{A}$ 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 $\mathcal{A}$ has an identity element^{} $e$, then clearly $e$ itself is an approximate identity for $\mathcal{A}$.
Title  approximate identity 

Canonical name  ApproximateIdentity 
Date of creation  20130322 17:30:22 
Last modified on  20130322 17:30:22 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  6 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46H05 
Synonym  approximate unit 
Defines  left approximate identity 
Defines  right approximate identity 