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[parent] $L^1(G)$ has an approximate identity (Theorem)

Let $ G$ be a locally compact topological group. In general, the Banach *-algebra $ L^1(G)$ (parent entry) does not have an identity element. In fact:

Proposition - $ L^1(G)$ has an identity element if and only if $ G$ is discrete.

When $ G$ is discrete the identity element of $ L^1(G)$ is just the Dirac delta, i.e. the function that takes the value $ 1$ on the identity element of $ G$ and vanishes everywhere else.

Nevertheless, $ L^1(G)$ has always an approximate identity.

Theorem - $ L^1(G)$ has an approximate identity $ (e_{\lambda})_{\lambda \in \Lambda}$. Moreover the approximate identity $ (e_{\lambda})_{\lambda \in \Lambda}$ can be chosen to satisfy the following properties:

where $ C_c(G)$ stands for the space of continuous functions $ G \longrightarrow \mathbb{C}$ with compact support.



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Also defines:  $L^1(G)$ has an identity element iff $G$ is discrete

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Cross-references: support, compact, continuous functions, approximate identity, vanishes, function, discrete, identity element, Banach *-algebra, topological group, locally compact

This is version 6 of $L^1(G)$ has an approximate identity, born on 2007-12-22, modified 2007-12-26.
Object id is 10155, canonical name is L1GHasAnApproximateIdentity.
Accessed 336 times total.

Classification:
AMS MSC: 22A10 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Analysis on general topological groups)
 22D05 (Topological groups, Lie groups :: Locally compact groups and their algebras :: General properties and structure of locally compact groups)
 43A20 (Abstract harmonic analysis :: $L^1$-algebras on groups, semigroups, etc.)
 46K05 (Functional analysis :: Topological algebras with an involution :: General theory of topological algebras with involution)
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