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 $A$.