Quoting from Timmermann's *[An invitation to quantum groups and duality](https://books.google.ie/books/about/An_Invitation_to_Quantum_Groups_and_Dual.html?id=gTeXAQAACAAJ&redir_esc=y)*:



> 
> **Prop. 5.1.3** Let $A$ 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?
> 
> 
> 



Showing a product on a character space is continuous
As per Nik Weaver's comment, this is a simple consequence of the fact that for compact Hausdorff spaces $X$, $Y$, for every unital \*-homomorphism $\pi:C(X)\rightarrow C(Y)$, there exists a continuous function $\phi:Y\rightarrow X$ such that, for $f\in C(X)$:
$$\pi(f)=f\circ \phi.$$


A reference for this fact is given by Nik's text, Mathematical Quantisation. 


