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Let $\Gamma$ be a discrete group (though this could be asked for general locally compact groups) and consider the Banach $*$-algebra $\ell^1(\Gamma)$. We have two natural $C^*$-algebra completions: the reduced group $C^*$-algebra $C^*_r(\Gamma)$ which is the closure of $\ell^1(\Gamma)$ acting by left translation on $\ell^2(\Gamma)$; and the full group $C^*$-algebra $C^*(\Gamma)$ which is the universal C$^*$-algebra completion of $\ell^1(\Gamma)$.

My (extremely hazy) intuition is that while $C^*(\Gamma)$ is the maximal $C^*$-completion, also $C^*_r(\Gamma)$ is the "minimal" one. But I can't really justify this. Is it right? To be precise:

Let $\pi:\ell^1(\Gamma)\rightarrow B(H)$ be an injective $*$-homomorphism, for some Hilbert space $H$. It is true that $\|\pi(a)\| \geq \|a\|_{C^*_r(\Gamma)}$ for all $a\in\ell^1(\Gamma)$??

For Abelian groups, I think some Fourier analysis shows this to be true. But what about in general?

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  • $\begingroup$ What about if in place of $\ell^1(\Gamma)$, we just take the group algebra $\mathbb C[\Gamma]$. Can there be a smaller completion in that case? $\endgroup$
    – Aaron Tikuisis
    Commented Jun 26, 2014 at 21:04
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    $\begingroup$ Aaron, there's no minimal C*-norm on $\mathbb{C}[\Gamma]$ in general. Take for example $\Gamma=\mathbb{Z}.$ Let $I_n$ be a decreasing sequence of subarcs of the circle with intersection a single point $\omega.$ The representations (call them $\pi_n$) corresponding to restriction to the $I_n$ are all injective on $\mathbb{C}[\mathbb{Z}].$ But any polynomial $p$ that has $\omega$ as a root will have $||\pi_n(p)||\rightarrow p(\omega)=0.$ $\endgroup$
    – Caleb Eckhardt
    Commented Jun 26, 2014 at 22:01

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The following is not an answer, but records some background and some related results which suggest that the original question might be hard to answer in the generality stated.

If $\Gamma$ is amenable, $\newcommand{\Cst}{{\rm C}^*}$ then the maximal and reduced $\Cst$ algebras coincide, and your question is equivalent to the following:

Question. Is the maximal $\Cst$ norm the unique $\Cst$ norm on $\ell^1(\Gamma)$?

We could ask the same question for a general locally compact (amenable) group $G$, not just the discrete ones. Consulting my colleague's copy of Palmer volume 2 and doing some digging online, it turns out that the answer is yes for groups of polynomial growth, but "no" for certain solvable Lie groups, see

J. Boidol, Group algebras with a unique $C^{\ast}$-norm. J. Funct. Anal. 56 (1984), no. 2, 220--232. MR 86c:22006

Some related results (without proofs) and references are in Section 10.5 of Palmer vol. 2. It seems that to get positive results one has to do noncommutative versions of the argument you allude to in your question for the commutative case, only instead of dealing with regular function algebras on the dual group you have to look at a suitable theory for noncommutative star-ideals.

There is a later paper of Leung and Ng

C.-W. Leung, C.-K. Ng, Some permanence properties of $C^*$-unique groups. J. Funct. Anal. 210 (2004), no. 2, 376--390. MR 2005c:22014

in which, on page 2, the authors say that it is unknown whether or not a discrete amenable group must have a unique $\Cst$ norm on its $\ell^1$-group algebra.

That said: none of this seems to indicate whether your original question has a positive or negative answer for, say, the free group on two generators, or SL(3,Z). Perhaps one can do something with conjugation-type representations?

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