# $L^{1}(G)$ 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 $(e_{\lambda})_{\lambda\in\Lambda}$. Moreover the approximate identity $(e_{\lambda})_{\lambda\in\Lambda}$ can be chosen to the following :

• $e_{\lambda}$ is self-adjoint (http://planetmath.org/InvolutaryRing),

• $\|e_{\lambda}\|_{1}=1$,

• $e_{\lambda}\in C_{c}(G)$

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

Title $L^{1}(G)$ has an approximate identity L1GHasAnApproximateIdentity 2013-03-22 17:42:40 2013-03-22 17:42:40 asteroid (17536) asteroid (17536) 9 asteroid (17536) Theorem msc 46K05 msc 43A20 msc 22D05 msc 22A10 $L^{1}(G)$ has an identity element iff $G$ is discrete