Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a *representation* of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that
$$
\pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\cdot\pi(b),\qquad a,b\in G.
$$
Consider the group ring, or, what is the same, the *group algebra* ${\mathbb C}[G]$ of $G$ over ${\mathbb C}$, and let $\delta:G\to{\mathbb C}[G]$ be the corresponding embedding (which is, of course, a representation of $G$).

It is obvious that every representation $\pi:G\to A$ can be (uniquely) extended to a homomorphism of algebras $\varphi:{\mathbb C}[G]\to A$: $$ \pi=\varphi\circ\delta, $$ (and vise versa, every such $\varphi$ generates $\pi$).

Moreover, this characterizes the group algebra ${\mathbb C}[G]$:

If $\delta:G\to B$ is a representation with the same property, then $B\cong {\mathbb C}[G]$.

This is what is called *group algebra* in Algebra. In Analysis the situation becomes completely different. The algebras playing the role of "classical" group algebras of topological groups, like $L^1(G)$, or $C^*(G)$, or $W^*(G)$ seem to do not have characterizations like that.

Am I right?

Are there any constructions of "group algebras" in Analysis (for some classes of topological groups $G$) that can be caracterized by this (or similar) universality property (so that they indeed have a right to be called "group algebras")?

The only examples that come to me are group algebras from the stereotype theory: ${\mathcal C}^\star(G)$, ${\mathcal E}^\star(G)$, ${\mathcal O}^\star(G)$, ${\mathcal R}^\star(G)$ (in these constructions the homomorphisms $\varphi$ must be continuous, and the representations $\pi$ must be continuous, smooth, holomorphic, regular, respectively -- see Theorem 10.12 here).

Is it possible that I miss something? Yemon Choi inspired me some doubts in our discussion here.