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.

$\|e_{\lambda}\|<C\;\;\;\;\forall_{\lambda\in\Lambda}\;$ , 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 :

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.

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	2013-03-22 17:30:22
Last modified on	2013-03-22 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