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?