Morphisms between compact quantum groups

Question

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.

Answer 1

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 $C^*_r(\Lambda)$.