Rieffel has given in his Ph.D. thesis a fairly satisfactory characterisation of commutative $L_1(G)$-algebras. This requires some terminology.

Let $A$ be a commutative (possibly non-unital) Banach algebra and let $f\in A^*$ be a (norm-one) character. Then $f$ is termed $L^\prime$*-inducing*, when $A$ is an abstract $L$-space under the ordering given by the cone

$$\{x\in A\colon \|x\| = f(x)\}$$

and $| x\cdot y | \leqslant |x| \cdot |y|$ in the sense of the above-introduced ordering (here $|x|$ stands for the Banach-lattice modulus). Rieffel then offers the following characterisation:

**Theorem.** Let $A$ be a commutative, semi-simple Banach algebra. Suppose that every norm-one character on $A$ is $L^\prime$-inducing and $A$ is Tauberian. Then $A$ is Banach-algebra isometrically isomorphic to $L_1(G)$ for some locally compact group $G$.

It is to be noted that there exist characterisations of $L_1(S)$-algebras, where $S$ is a locally compact semigroup, that are in a spirit similar to the above result. There is a recent (2016) result by Lau and Hung extending this further to general Fourier-Stieltjes algebras.