Let $G$ be an Abelian Locally Compact Group. In Representation Theory one usually looks at the $L^1 (G)$ convolution algebra, which together with the convolution operation and $L^1$ norm is a commutative Banach $*$ Algebra.
One of the fundamental relations between the representation theory of $G$ and $L^1(G)$ is that the multiplicative characters of $G$ correspond directly to multiplicative functionals on $L^1(G)$ (which are also called characters): for a character $\xi: G \to \mathbb{T}$ one defines a functional on $L^1(G)$ by $\xi(f) = \int \xi (g) f(g)dg$ (w.r.t Haar Measure), and all multiplicative functionals arise this way. So $\sigma(L^1(G)) = \widehat{G}$.
However, one can also look at the algebra $M(G)$ of all complex (Radon) measures on $G$, together with convolution of measures $\mu * \nu$ and the total variation norm, this is commutative Banach Algebra (one can also think of notions of involution on it). Each $\xi$ again defines a functional $\xi(\mu) = \int \xi d\mu$ which is again multiplicative, and so we can write $\widehat{G} \hookrightarrow \sigma(\mathcal{M}(G))$.
My question is, how can we fully describe $\sigma(\mathcal{M}(G))$? Is there anything else? Is it of any interest? Any kind of helo is appreciated.
My Attempt: Somehow I "feel" that we should have $\widehat{G} = \sigma(\mathcal{M}(G))$, since multiplicative functionals on a bigger algebra restrict to ones on the smaller one. But $L^1(G) \subset \mathcal{M}(G)$ shouldn't be dense, yet how do we extended characters? I would intuitively guess there are more characters on a richer algebra.
