On convolution products and automorphisms in Hopf \(C^*\)-algebras (Q740833)

The author considers the bi-inner Hopf \(*\)-automorphisms of a Hopf \(C^*\)-algebra, that is, those Hopf \(*\)-automorphisms that are inner as algebra automorphism, both in the dual algebra and in the given algebra. Two characterizations of the bi-inner Hopf \(*\)-automorphisms for the case of finite-dimensional Hopf \(C^*\)-algebras are given. Then it follows that the set of bi-inner Hopf \(*\)-automorphisms of a finite-dimensional Hopf \(C^*\)-algebra is a connected Lie group. Some interesting techniques involving positivity-preserving maps, faithful linear functionals and other related concepts are developed.