${L}^{1}(G)$ has an approximate identity
Let $G$ be a locally compact topological group. In general, the Banach *algebra ${L}^{1}(G)$ (parent entry (http://planetmath.org/L1GIsABanachAlgebra)) does not have an identity element. In fact:
 ${L}^{1}(G)$ 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 ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$. Moreover the approximate identity ${({e}_{\lambda})}_{\lambda \in \mathrm{\Lambda}}$ can be chosen to the following :

•
${e}_{\lambda}$ is selfadjoint (http://planetmath.org/InvolutaryRing),

•
${\parallel {e}_{\lambda}\parallel}_{1}=1$,

•
${e}_{\lambda}\in {C}_{c}(G)$
where ${C}_{c}(G)$ stands for the space of continuous functions $G\u27f6\u2102$ with compact support.
Title  ${L}^{1}(G)$ has an approximate identity 

Canonical name  L1GHasAnApproximateIdentity 
Date of creation  20130322 17:42:40 
Last modified on  20130322 17:42:40 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  9 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46K05 
Classification  msc 43A20 
Classification  msc 22D05 
Classification  msc 22A10 
Defines  ${L}^{1}(G)$ has an identity element iff $G$ is discrete 