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:

• $e_{\lambda}$ is self-adjoint,
• $\|e_\lambda\|_1 = 1$ ,
• $e_{\lambda} \in C_c(G)$