Let $(A, \Delta)$ be a compact quantum group in the sense of Woronowicz. Is it true that the comultiplication $\Delta : A \to A \otimes A$ always injective?


This is true for both the universal (because one has a counit) and the reduced (because the Haar state is faithful) version, but the general case seems more delicate.


In particular, is it true for the dual of a discrete group $\Gamma$ realized as a compact $C^\*$-algebraic quantum group using a $C^\*$-norm that lies strictly between the universal and the reduced $C^\*$-norms? What about the case where $\Gamma$ is the free group on two generators?



Is the comultiplication of a compact quantum group always injective?
No, the comultiplication need not be injective. When $\Gamma$ is a countable group and $\pi : \Gamma \to \mathcal{U}(H)$ is a faithful unitary representation with the property that $\pi \otimes \pi$ is weakly contained in $\pi$, we write $A = C^\*\_\pi(\Gamma)$ and there is a unique unital $\*$-homomorphism $\Delta : A \to A \otimes A$ satisfying $\Delta(\pi(g)) = \pi(g) \otimes \pi(g)$ for all $g \in \Gamma$. Then, $(A,\Delta)$ is a compact quantum group in the sense of Woronowicz.


Now $\Delta$ is faithful if and only if $\pi$ is weakly contained in $\pi \otimes \pi$. That need not be the case, as the following example with $\Gamma = \mathbb{F}\_2$ shows.


For every $0 < \rho < 1$, we consider the function $\varphi\_\rho : \mathbb{F}\_2 \to \mathbb{R}$ given by $\varphi\_\rho(g) = \rho^{|g|}$, where we use the word length $|g|$. By [Haa, Lemma 1.2], $\varphi\_\rho$ is a positive definite function and we define $\pi\_\rho$ as the associated cyclic representation. Denote by $\lambda$ the regular representation. By [Haa, Theorem 3.1], we have that $\pi\_\rho$ is weakly contained in $\lambda$ if and only if $\rho \leq 1/\sqrt{3}$.


We now claim that with $\rho = 2/3$, the representation $\pi = \pi\_\rho \oplus \lambda$ provides a counterexample for the faithfulness of $\Delta$. By construction, $\pi \otimes \pi$ is weakly equivalent with $(\pi\_\rho \otimes \pi\_\rho) \oplus \lambda$. Since $\rho^2 < 1/\sqrt{3}$, it follows from [Haa, Theorem 3.1] that $(\pi \otimes \pi) \sim \lambda \prec \pi$. But since $\rho > 1/\sqrt{3}$, we have $\pi\_\rho \not\prec \lambda$, so that $\pi \not\prec \pi \otimes \pi$.


[Haa] U. Haagerup, An example of a nonnuclear C$^\*$-algebra, which has the metric approximation property. Invent. Math. 50 (1978/79), 279-293.


