# approximate identity

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 :

1. 1.

$\|e_{\lambda}\|, for some constant $C$.

2. 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 :

1. 1.

$\|e_{\lambda}\|, for some constant $C$.

2. 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.

## 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 ApproximateIdentity 2013-03-22 17:30:22 2013-03-22 17:30:22 asteroid (17536) asteroid (17536) 6 asteroid (17536) Definition msc 46H05 approximate unit left approximate identity right approximate identity