Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital $*$-homomorphism, which is coassociative (meaning that $(\Delta \otimes \operatorname{id})\Delta-(\operatorname{id} \otimes \Delta)\Delta$) and the following cancelation laws are satisfied: $\Delta(A)(1 \otimes A)$ and $\Delta(A \otimes 1)$ are linearly dense in $A$. Such objects have well behaved representation theory: first let me just quickly recall what is a (finite dimensional) representation in this context. In is simply the matrix $u \in M_n(A)$ for some $n$ such that $\Delta(u_{ij})=\sum_k u_{ik} \otimes u_{kj}$. One can show that there are sufficiently many such representations: if we put $A_0$ to be the space spanned by coefficients of (finite dimensional, unitary) representations than we find that $A_0$ equipped with $\varepsilon(u_{ij})=\delta_{ij}$ and $S(u_{ij})=u_{ji}^*$ becomes a Hopf $*$-subalgebra of $A$, which is dense. My question is:

How to prove that the linear span of $\Delta(A_0)(1 \otimes A_0)$ (where this time tensor product is the algebraic one) is whole $A_0 \otimes A_0$?


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