Let $\mathbb{G}= (A, \Delta)$ be a ($C^\*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^\*= B(A, \mathbb{C})$ obtains a product  

$$\omega\_1\*\omega\_2:= (\omega\_1\otimes \omega\_2) \circ \Delta$$
and this is used to prove the existence of the Haar functional on a compact quantum group.


**Question:** How is $\omega\_1 \otimes \omega\_2$ defined here? Clearly we have a linear mapping $$\omega\_1 \odot \omega\_2: A \odot A \to \mathbb{C}$$
on the algebraic tensor product, but we need continuity to extend this to the completion $A \otimes A$ (with respect to the minimal $C^\*$-norm on the algebraic tensor product $A \odot A$).


In general, I believe $\omega\_1 \odot \omega\_2$ must not be continuous, though this result does hold when one works with states on the $C^\*$-algebra $A$.



Convolution of functionals on compact quantum group
Every bounded linear functional on a $C^\ast$-algebra is a linear combination of states, so $\omega\_1\odot \omega\_2$ extends to the spatial (minimal) tensor product for all $\omega\_1, \omega\_2 \in A^\ast$ by a theorem of Takesaki.


