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](http://en.wikipedia.org/wiki/Group_algebra) 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:


1. 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 $G$?
2. 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.



Which Banach algebras are group algebras?
A characterization of the Banach algebras which are isometrically \*-isomorphic to $L^1(G)$ for some $G$ was given by P. L. Patterson, Characterization of algebras arizing from locally compact groups, Trans. Amer. Math. Soc. 329 (1992), 489-506. This paper contains references to earlier ones dealing with special classes of groups, in particular Abelian and compact ones. 


