$L^1(G)$ 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 self-adjoint (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 $G⟶ℂ$ with compact support.