Let $\mathbb{G}$ be a $C^\*$-algebraic compact quantum group. Consider the associated dense Hopf$^\*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ denote its antipode. In the book "Compact quantum groups and their representation categories", it is shown that $S$ is unbounded when $\mathbb{G}$ is not of Kac-type. The converse is also true, namely if $\mathbb{G}$ is of Kac-type, then $S$ is bounded. To see this, note that the assumption implies that $S$ is $\*$-preserving, and thus $S$ is a positive map because
$$S(x^\*x) = S(x)S(x^\*) = S(x)S(x)^\*\ge 0$$
for any $x \in \mathcal{O}(\mathbb{G})$. Note that $\mathcal{O}(\mathbb{G})$ is an operator system, and since a positive unital map on an operator system is necessarily bounded (See Paulsen's book "Completely bounded maps and operator algebras", proposition 2.1), we conclude that $S$ is bounded. Moreover, it also follows that $\|S\| \le 2\|S(1)\| = 2.$


**Question**: Can we say something more about the norm $\|S\|$? Is it possible that $\|S\| = 2?$



Norm antipode on a Kac-type compact quantum group
$S$ is an anti-$\*$-homomorphism, and extends by continuity to $A = C(\mathbb G)$ (the closure of $\mathcal O(\mathbb G)$ acting on the GNS space for the Haar state). Let $A^{\operatorname{op}}$ be the opposite $C^\*$-algebra to $A$. Then we can consider $S$ as a map $A\rightarrow A^{\operatorname{op}}$, which is now a $\*$-homomorphism, and hence contractive.


The relevant result in the book of Neshveyev and Tuset is Proposition 1.7.9. This also shows that (equivalently) $S^2=\operatorname{id}$. So in fact $S$ is an isometry.


