Quoting from Timmermann's *An invitation to quantum groups and duality*:

Prop. 5.1.3Let $A$ be a commutative algebra of functions on a compact quantum group. Then there exists a compact group $G$ and an isomorphism $A\cong C(G)$ of $\mathrm{C}^*$-algebras.

Proof:By the Gelfand theorem, there exists a compact space $G$, a continuous map $m:G\times G\rightarrow G$...

I might reduce this to:

Let $A$ be a commutative unital $\mathrm{C}^*$-algebra with a $\mathrm{C}^*$-morphism $\Delta:A\rightarrow A\otimes A$. By the Gelfand theorem, where $\Phi(A)$ is the Hausdorff weak*-compact character space, the map $m:\Phi(A)\times \Phi(A)\rightarrow \Phi(A)$, $(\chi,\varphi)\mapsto (\chi\otimes\varphi)\Delta$ is continuous...

My difficulties are more appropriate to MSE, although the content knowledge might be more appropriate to here.

My natural inclination is to take sequences $(\chi_n)\rightarrow \chi \in \Phi(A)$ and $(\varphi_n)\rightarrow\varphi\in\Phi(A)$ and to look at $(\chi_n,\varphi_n)$, and show: $$m(\chi_n,\varphi_n)\rightarrow m(\chi,\varphi),$$ and using the dense subalgebra generated by the matrix elements of unitary irreducible (co)representations $(\rho_{ij}^\alpha)_{\alpha\in\text{Irr}(A)}$ shows: $$(\chi_n\otimes\varphi_n)\Delta(\rho_{ij}^\alpha)\rightarrow (\chi\otimes\varphi)\Delta(\rho_{ij}^\alpha),$$

but I don't think that $\Phi(A)$ is in general a sequential space. I am happy that it is when $A$ is separable, but I understand that the opening proposition is believed to be true in the non-separable case also.

Woronowicz originally assumed that the algebra of functions on a compact quantum group was separable in order to deduce the existence of a Haar state. Van Daele removed this assumption.

How can we show that the map $\Phi(A)\times \Phi(A)\rightarrow \Phi(A)$ is continuous when $A$ is unital, commutative, non-separable?