About the commutativity of the algebra $L^1 (G)$

Question

I would like to know, under what condition on the group $G$ (abelian, compact or localement compact ...), the algebra $L^1 (G)$ is commutative?

Thank you

Answer 1

For a locally compact group $G$, the convolution algebra $L^1(G)$ is commutative if and only if G is Abelian.

This is Theorem 1.6.4 in Principles of Harmonic Analysis by Deitmar and Echterhoff.