Which group algebras in analysis are "true group algebras"? 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.
 A: Just to avoid prolonged discussion in comments I'll put a partial answer for the discrete case here — at the time of writing, I am less sure about the precise picture for non-discrete groups, although some parts carry over since one can use the Bochner integral to convert suitable continuous representations $\pi:G\to A$ to continuous algebra homomorphisms $L^1(G)\to A$.
So, let $G$ be a group (considered as having the discrete topology). Then every bounded representation of $G$ in a unital Banach algebra $A$ extends uniquely to a unital algebra homomorphism $\ell^1(G)\to A$.
As Tobias Fritz has observed in comments: every $*$-representation of $G$ in a unital ${\rm C}^*$-algebra $A$ extends uniquely to a unital $*$-homomorphism ${\rm C}^*(G)\to A$, and since the original representation must map $G$ into a subgroup of ${\mathcal U}(A)$, one can indeed view the universal property as showing ${\rm C}^*$ is a left adjoint to a suitable forgetful functor (by the "initial object" description of left adjoints. (For locally compact groups that are non-discrete, I think Ernest's $W^*(G)=C^*(G)^{**}$ has an analogous description but where one uses von Neumann algebras and normal $*$-homomorphisms as the target category; however I have not checked the details.)
It is less obvious how $\ell^1(G)$ might be viewed as a left adjoint since general Banach algebras, since the natural analogues of the "unitary group" do not have such good properties as in the ${\rm C}^*$-world.
 However, I claim that the $\ell^1$-monoid algebra construction (which when applied to a monoid that happens to be a group) does have a reasonable interpretation as a left adjoint).
In detail: let us define a representation of a monoid $S$ in a (Banach) algebra $A$ as being a function $\pi:S \to A$ satisfying $\pi(e_S)=1_A$ and $\pi(st)=\pi(s)\pi(t)$ for all $s,t\in S$. Let ${\rm ball}(A)$ denote the closed unit ball of a Banach algebra $A$, so that if $A$ is unital then ${\rm ball}(A)$ is a monoid. Then ${\rm ball}$ is a functor from the category of unital Banach algebras and unital homomorphisms of norm $\leq 1$ to the category of monoids and monoid homomorphisms; and the left adjoint to this functor is the $\ell^1$-monoid algebra.
