A Peter–Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions

Abstract

We show that a relatively ergodic extension of measure-preserving dynamical systems has relative discrete spectrum if and only if it can be represented as a skew-product by a bundle of compact homogeneous spaces. Our result holds without restrictions on the acting group or the underlying probability spaces. This generalizes previous work by Mackey, Zimmer, Ellis, Austin, and the second author and Tao, and is inspired by the Furstenberg–Zimmer and Host–Kra structure theories for actions of uncountable groups. Our approach uses a natural model to answer the ergodic-theoretic question with the help of structure theory for topological dynamical systems. A key step in our argument is establishing a Peter–Weyl-type theorem for bundles of compact groups which might be of independent interest.