As is well known, using the Hilbert Nullstellensatz (and a more recent result of Cartier) one can show that commutative finitely generated Hopf algebras over $\mathbb{C}$ are equivalent to algebraic groups (cf the accepted answer of David Speyer). What happens when one considers finitely generated Hopf $*$-algebras? Does there exist an analogous equivalence with the star coming (perhaps) from matrix adjoints?
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I think you have to assume that your commutative Hopf $*$-algebra is spanned by the matrix coefficients of unitary corepresentations (in which case that's the case: [1] or [2, Theorem 1.6.7] to get Woronowicz compact quantum group and one can use the Gelfand-Naimark duality). For example, take $H$ to be the group algebra of $\mathbb{Z}/3$, endowed with the nonstandard $*$-structure $\alpha g \mapsto \bar{\alpha} g$ for any $g\in\mathbb{Z}/3$. Then the irreducible corepresentations of $H$ are not unitarizable.
[1] Mathijs S. Dijkhuizen and Tom H. Koornwinder, CQG algebras: a direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32 (1994), no. 4, 315–330. MR 1310296 (95m:16029) [2] Sergey Neshveyev and Lars Tuset, MR 3204665 Compact quantum groups and their representation categories, ISBN: 978-2-85629-777-3.
