Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ mathfrak{g} $ ) associated to $ mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ mathcal{G} $ . When the underlying manifold of $ mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{mathfrak{g}}_{mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.