# What is the coproduct in the category of commutative unital C*-Algebras?

Let $A$ be a commutative unital C*-Algebra and let $X= Spec(A)$ be the corresponding compact Hausdorff space of characters. By Gelfand-Naimark duality we know that $$X \times X = Spec(A \coprod A)$$ or in other words $$A \coprod A = \mathcal{C}(X \times X).$$

Is there an algebraic way to construct the coproduct, as for example some form of topological tensor product or similar?

• The theorem stated in the question here might be a good rule of thumb. My guess is the tensor product (which is unambiguous since commutative C$^*$-algebras are nuclear). Maybe check whether the tensor product is a coproduct. – Josh Keneda Sep 5 '16 at 12:03

Let $A, B,$ and $C$ be C$^*$-algebras. If $\phi: A \rightarrow C$ and $\psi: B \rightarrow C$ are $*$-homomorphisms whose images commute, then there's a unique $*$-homomorphism $\phi \otimes \psi : A\otimes_{max} B \rightarrow C$ such that $(\phi\otimes \psi) (a \otimes b) = \phi(a)\psi(b)$. See page 193 of Murphy's book for a proof here.
If we take $A \otimes_{max} B$ with inclusions $i_A (a) = a \otimes 1$ and $i_B(b) = 1 \otimes b$, then the universal property above guarantees that we have found our coproduct in C$^*$-alg$_{com}^1$. Note that the "max" in the previous sentence wasn't necessary, since $A$ and $B$ are nuclear anyway. So you could just as easily take the spatial tensor product, if you like that construction better.