Compact quantum group

Remarks

Example: let $G$ be a compact Hausdorff group and $A = C(G)$, so that $A \otimes A = C(G \times G)$. Set $\Delta(f)(s, t) = f(st)$. Coassociativity is the associativity $(rs)t = r(st)$. The products $\Delta(f)(1 \otimes g)$ are the functions $(s, t) \mapsto f(st) g(t)$. Since $(s, t) \mapsto (st, t)$ is a homeomorphism of $G \times G$, their span is a self-adjoint subalgebra that separates points and contains $1$. So it is dense by the Stone–Weierstrass theorem, and the other span is handled in the same way.

Non-example: let $S = [0, 1]$ with ordinary multiplication, a compact semigroup that is not a group, and define $\Delta$ on $C(S)$ by the same formula. Clause 1 holds, but every function $(s, t) \mapsto f(st) g(t)$ is constant in $s$ on the line $t = 0$. This property passes to linear combinations and uniform limits, so the span is not dense, and clause 2 fails.

For commutative $A$, every compact quantum group is of the form $C(G)$ for a compact group $G$. Every compact quantum group has a unique state $h$, the Haar state, with $(h \otimes \id)\Delta(a) = (\id \otimes h)\Delta(a) = h(a) 1$ for all $a \in A$; it plays the role of the Haar measure.