Let $(A, \Delta\_A)$ and $(B, \Delta\_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta\_A) \to (B, \Delta\_B)$ to be a unital $\*$-morphism
$$\pi: B \to A$$
such that $$(\pi \otimes \pi)\circ \Delta\_B = \Delta\_A \circ \pi.$$


However, in the literature, I read that this definition is not the 'right' one because there are analytical subtleties. Instead, such a morphism is defined to be a $\*$-morphism $\pi: B\_0 \to A\_0$ satisfying $(\pi \otimes \pi)\Delta\_B = \Delta\_A \pi$ where $A\_0$ and $B\_0$ are the associated spaces of matrix coefficients of finite-dimensional unitary representations. Equivalently , we can translate this to a $\*$-morphism $B\_u \to A\_u$ between the associated universal compact quantum groups in the sense that I proposed above.


What can go wrong with the definition I propose? Why is the definition with matrix coefficients the 'correct one'?


Also, for most purposes, having a $\*$-morphism $B\to A$ respecting the comultiplications seems good enough, so I am confused why one chooses to work with the matrix coefficients instead. For practical purposes, this seems bad as one has to know a lot of information about the spaces of matrix coefficients involved.



Morphisms between compact quantum groups
When $\pi : B \to A$ is a unital $\*$-homomorphism respecting the comultiplication, then automatically $\pi(B\_0) \subseteq A\_0$, because $\pi$ maps a corepresentation of $(B,\Delta\_B)$ to a corepresentation of $(A,\Delta\_A)$.


The converse need not hold: if $\pi : B\_0 \to A\_0$ is a unital $\*$-homomorphism respecting the comultiplication, it need not be bounded for the C$^\*$-norms on $B$ and $A$. So, by only considering $\*$-homomorphisms $B \to A$, you may ``miss'' some quantum group morphisms. The required C$^\*$-boundedness would be automatic if $B=B\_u$ is the universal enveloping C$^\*$-algebra of $B\_0$.


All this is best illustrated when $(B,\Delta\_B)$ and $(A,\Delta\_A)$ are C$^\*$-algebras of discrete groups $\Lambda$ and $\Gamma$. Every group homomorphism $\delta : \Lambda \to \Gamma$ gives rise to a natural $\*$-homomorphism $C^\*(\Lambda) \to C^\*(\Gamma)$, but not necessarily from $C^\*\_r(\Gamma)$ to $C^\*\_r(\Lambda)$.


