I am studying Alain Robert's Introduction to the Representation Theory of Compact and Locally Compact Groups, where I found a notion that might be generalized.
Given a locally compact group $G$ and a representation on a Banach space $V$ by $\pi: G \to GL(V)$, we have an induced representation of algebra $\pi^1: L^1(G) \to End(V)$ by integration of representation $\pi^1(f) = \int_G f(g)\pi (g) \,dg$. (Here, $L^1(G)$ is an algebra under convolution.)
The author mentioned that although $G$ cannot be embedded in $L^1(G)$ unless $G$ is discrete, we can still think $L^1(G)$ as an analogue of group algebra.
What I have in mind is: If we really want the real analogue of group algebra, why don't we consider, $D(G)$, the space of (perhaps tempered) distributions? After all, what we want are the Dirac delta functions!
I wonder if there is any trouble to use $D(G)$, or it is just that the author did not want to get into the most general theory.
Speaking of general theory.. I have some more questions on this topic.
How general can the vector space be? The author mentioned in his book that $V$ can more generally be "quasi-complete". Is this the most general situation, and is there any good reason for this (if any)?
When $G$ is finite, we can recover $Rep(G)$ from $Rep(\mathbb{C}[G])$. Is is the case that we can recover $Rep(G)$ when G is merely locally compact? My guess is we can recover it from $Rep(D(G))$.