On square roots of the uniform distribution on compact groups

Abstract

Let G G be a compact separable topological group. When does there exist a probability P P such that P ∗ P = U P * P = U , where U U is Haar measure and P ≠ U P \ne U ? We show that such square roots exist if and only if G G is not abelian, nor the product of the quaternions and a product of two element groups. In the course of proving this we classify compact groups with the property that every closed subgroup is normal.