Positive definite superfunctions and unitary representations of Lie supergroups (Q367154)

The authors show that, for a rather big class of Fréchet-Lie supergroups (which is characterized by a condition on the convergence of the Trotter product formula), there is a correspondence between positive definite smooth or analytic superfunctions and matrix coefficients of certain smooth and analytic unitary representations, respectively. This generalizes a well-known result for some finite dimensional supergroups, however, in the infinite dimensional case the proof requires various new ideas. As an application, it is shown that a smooth positive definite superfunction is analytic if and only if it restricts to an analytic function on the underlying manifold of the supergroup. When the underlying manifold is \(1\)-connected, the authors obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra to correspond to a matrix coefficient of a unitary representation.