# 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.

• You seem to be asking if there are "initial object" characterizations of $L^1(G)$ and $C^*(G)$ in appropriate categories of topological algebras and continuous homomorphisms. Is that correct, or have I misunderstood? – Yemon Choi Feb 7 '15 at 12:30
• Moreover: if $G$ is locally compact and non-discrete then $L^1(G)$ and $C^*(G)$ are not unital algebras, so they cannot be initial objects in categories of "homomorphisms from $G$ to unital algebras". How important is it to you that $\varphi: G\to A$ is a unital homomorphism? Would you be happy if there was a way to characterize $\ell^1(G)$ and $C^*(G)$ for $G$ discrete via some universal property? – Yemon Choi Feb 7 '15 at 12:45
• At least the maximal group C*-algebra $C^*(G)$ of a discrete group $G$ does have a well-known universal property: just as $\mathbb{C}[G]$ classifies ordinary representations of $G$, $C^*(G)$ classifies unitary representations, in the sense that every group homomorphism $\pi:G\to \mathcal{U}(B)$ to the unitary group $\mathcal{U}(B)$ to another C*-algebra $B$ uniquely extends to a $*$-homomorphism $C^*(G)\to B$. More concisely speaking, the functor $G\mapsto C^*(G)$ which assigns to every discrete group its maximal group C*-algebra is right adjoint to the unitary group functor. – Tobias Fritz Feb 7 '15 at 12:53
• @TobiasFritz Yes this is well known and this is why I wondered if SergeiAkbarov had something else in mind. If $G$ is discrete then $\ell^1(G)$ has a similar universal property with respect to bounded unital homomorphisms into arbitrary Banach algebras. – Yemon Choi Feb 7 '15 at 12:55
• Yemon, yes if $L_1(G)$ or $C^∗(G)$ could be characterized as initial objects, that would be interesting. I suppose, you mean initial object in some category of representations of G''? I did not understand, are $C^∗(G)$ or $L_1(G)$ such initial objects for discrete $G$ or for some wider classes of groups? – Sergei Akbarov Feb 26 '18 at 6:30

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.

• It might be worthwhile mentioning that the "suitable forgetful functor" is the one which assigns to every C*-algebra $A$ its unitary group $\mathcal{U}(A)$, considered as a discrete group. – Tobias Fritz Feb 7 '15 at 13:57
• So as far as I understand, for the case of discrete $G$ you and Tobias suggest two examples: 1) $C^*(G)$ will be an initial object in the category of unitary representations of $G$ in C*-algebras, and 2) $\ell^1(G)$ will be an initial object in the category of representations of $G$ in Banach algebras. Is it correct? – Sergei Akbarov Feb 7 '15 at 14:01
• @SergeiAkbarov Yes that is correct: in the Banach case one needs to restrict to bounded representations – Yemon Choi Feb 7 '15 at 14:12
• Ah, OK! That's interesting... What is the problem in the non-discrete case? – Sergei Akbarov Feb 7 '15 at 14:13
• And are there other constructions? Will $W^*(G)$ be initial object anywhere? – Sergei Akbarov Feb 7 '15 at 14:19