Coproduct of $C^*$-algebras I want to prove that the free product $A*B$ of two unital $C^*$-algebras $A$ and $B$ is a coproduct in the sense of category theory. Remember the construction of $A*B$: Take generators $\{a:a\in A\}\cup\{b:b\in B\}$ and let $A*B$ the $C^*$-algebras generated by this set. Then the inclusion maps $\iota_A$ and $\iota_B$ are obvious and are $*$-homomorphisms. Now let $X$ be another $C^*$-algebra with *-homomorphisms $\mu:A\rightarrow C$ and $\nu:B\rightarrow C$, then we can define $u:A*B\rightarrow C$ by $u(a)=\mu(a)$ and $u(b)=\nu(b)$, then this extends to a $*$-homomorphism $u:A*B\rightarrow C$. My question is: why is this map the unique map such that $u\circ\iota_A=\mu$ and $u\circ\iota_B=\nu$? Can someone help me?
Thank you very much.
 A: Since $u\circ i_A=\mu$, we have $u(i_A(a))=\mu(a)$ for all $a\in A$; hence $u(a)=a$ (if, as you do, you identify $A$ with its image $i(A)$). At the same time, $u(b)=\nu(b)$ for $b\in B$. Thus, $u$ is uniquely determined on $A\cup B\subset A*B$, which uniquely determines $u$ on the entire $A*B$. 
A: If I'm not mistaken to badly, algebraic set-indexed coproducts exist.
Let $*_\lambda A_\lambda$ be the coproduct in the category of *-algebras.
Take the supremum over all C*-seminorms $\delta:*_\lambda A_\lambda\to\mathbb{R}_+$ that agree with the original C*-norms, i.e. $\delta(a_\lambda)=\|a_\lambda\|_\lambda$ for all $a_\lambda\in A_\lambda$.
Note that a priori there may be no such C*-seminorm at all, but which wouldn't be an issue since in that case the supremum would give the trivial C*-seminorm. More importantly, however, is that it is finite for every element, which is luckily the case since
$$\delta(\sum_ka_kb_k\ldots)\leq\sum_k\delta(a_k)\delta(b_k)\ldots\leq\sum_k\|a_k\|_{\lambda(a_k)}\|b_k\|_{\lambda(b_k)}\ldots$$
which is bounded independently of the C*-seminorm.
Thus one may take the Hausdorff completion.
The universal property is then easily shown.
A: Héhé, I found this on internet, just a small paragraph on page 4 (or 246) talking about which norm one should take.
