I am dealing with two "universal completions" but I am not sure if they are the same thing and would appreciate some guidance.
- Let $\mathcal{A}$ be a unital *-algebra. A $\mathrm{C}^*$-seminorm on a $\mathcal{A}$ is a seminorm $p:\mathcal{A}\rightarrow \mathbb{R}^+$ that satisfies, for all $a,\,b\in\mathcal{A}$:
$$p(ab)\leq p(a)p(b)\text{ and }p(a^*a)=p(a)^2.$$
We get a 'large' (pre?)-$\mathrm{C}^*$-norm by taking:
$$a\mapsto \|a\|_{u_1}:=\sup\{p(a)\mid p\text{ is a $\mathrm{C}^*$-seminorm on $\mathcal{A}$}\}.$$ Denote $A_{u_1}$ the corresponding norm-completion of $\mathcal{A}$.
- $\mathcal{A}$ as before, for $H$ Hilbert spaces, another 'large' (pre?)-$\mathrm{C}^*$-norm:
$$a\mapsto \|a\|_{u_2}:=\sup\{\|\pi(a)\|\mid\pi:\mathcal{A}\rightarrow B(H),\text{ a unital *-homomorphism}\}.$$
Denote $A_{u_2}$ the corresponding norm-completion of $\mathcal{A}$.
Question: Are these two norms (and hence $\mathrm{C}^*$-algebras) the same thing?
Remark: The Hilbert space $H$ should vary as well when taking the supremum, otherwise the two can be different and you generally only have (depending on what $H$ you chose) $\|\cdot\|_{u_2}≤\|\cdot\|_{u_1}$.
Suppose $p$ is a non-zero $C^*$ semi-norm. The zero locus of $p$ is a two-sided $*$ ideal, as such $\mathcal A/N(p)$ is also a unital $*$-algebra, if we complete it then $p$ makes it into a $C^*$-algebra. By using for example the GNS construction you find a Hilbert space $H$ and an injective unital representation $\pi:\overline{\mathcal A/N(p)}\to B(H)$. Injective $*$-morphisms are isometric for $C^*$-algebras so you have $$p(a) = p([a]) = \|\pi([a])\|$$ so the representation $\mathcal A\to B(H)$, $a\mapsto \pi([a])$ is a unital isometric $*$ representation if you give $\mathcal A$ the $C^*$ semi-norm $p$.
Conclusion: Every $C^*$ semi-norm comes from a unital representation to operators on a Hilbert space, as such the two sets over which you take the supremum are the same.