Given a locally compact Hausdorff group $G$, one can construct several Banach star-algebras using $G$ (and its associated Haar measure): $L^1 (G)$, $M(G)$ (regular complex measures on $G$), $L^{\infty} (G)$, $C^* (G)$, $C^*_r (G)$, $W^* (G)$, etc. (see this Wiki article for some details). Then one can ask, for instance, which Banach *-algebras can be represented (i.e. are isometrically *-isomorphic) to/as $L^{\infty} (G)$. In this case every such algebra is an abelian Von-Neumann algebra, and every abelian Von-Neumann algebra can be represented as $L^ \infty (X,\mu)$ for some set $X$ with measure $\mu$`, so the question reduces to the question "which measures are Haar measures (on some LCH group)?". However, I'm mostly interested in this type of question for the algebras $L^1 (G)$. It is known that such a Banach *-algebra is semisimple and symmetric for any **abelian** LCH group $G$, so obviously one can't represent all commutative Banach *-algebras as $L^1 (G)$. So, to summarize, I have two questions:

Is there an operator-theoretic characterization of the Banach algebras which are isometrically *-isomorphic to $L^1 (G)$ for some $G$? In particular, is there an abelian, symmetric, semisimple Banach *-algebra which isn't $L^1 (G)$ for some $G$?

Are there classes of Banach *-algebras to which one can associate an LCH group $G$ in a

*canonical*way (up to an isomorphism of topological groups), so that the algebra is*naturally*isomorphic to a group algebra $L^1 (G)$?

Also, if there is some known reference for this kind of problems, I'll be glad to know of it.