PlanetMath: approximate identity

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approximate identity

(Definition)

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:

$\Vert e_{\lambda}\Vert < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$ , for some constant .
$e_{\lambda}a \longrightarrow a\;$ , 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:

$\Vert e_{\lambda}\Vert < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$ , for some constant .
$ae_{\lambda} \longrightarrow a\;$ , 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 has an identity element , then clearly itself is an approximate identity for $\mathcal{A}$ .

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"approximate identity" is owned by asteroid. [ full author list (2) ]

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Other names:

approximate unit

Also defines:

left approximate identity, right approximate identity

Attachments:: -algebras have approximate identities (Theorem) by asteroid

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Cross-references: identity element, net, Banach algebra
There are 7 references to this entry.

This is version 2 of approximate identity, born on 2007-08-25, modified 2007-08-25.
Object id is 9895, canonical name is ApproximateIdentity.
Accessed 1035 times total.

Classification:

AMS MSC:

46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras)

Pending Errata and Addenda

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