Reduction of structure groups for noncommutative principal bundles

Abstract

Let $G$ be a compact group and let $H \subseteq G$ be a closed subgroup. We study reduction of the structure group for noncommutative principal bundles, modelled by free C${}^*$-dynamical systems in the sense of Ellwood. We first describe how induction from $H$ to $G$ behaves on associated correspondences and their tensor structure. Using reconstruction, this leads to a Tannakian characterization of reduction of the structure group in terms of the associated tensor functor. For normal subgroups, we characterize reduction in terms of central equivariant homomorphisms and derive module- and $K$-theoretic obstructions. We conclude with a noncommutative $O(n)$-example which admits no reduction to $SO(n)$.