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

$$omega_1*omega_2:= (omega_1otimes omega_2) circ Delta$$

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 $A$.