Positive definite superfunctions and unitary representations of Lie supergroups

TL;DR

This paper establishes a correspondence between positive definite superfunctions and unitary representations of Lie supergroups, providing a new framework for understanding their structure and functional analysis.

Contribution

It introduces a novel correspondence for a broad class of Frechet-Lie supergroups linking superfunctions to unitary representations and characterizes related linear functionals.

Findings

Established a correspondence between positive definite superfunctions and unitary representations.

Characterized linear functionals on the universal enveloping algebra related to analytic unitary representations.

Extended the theory to a broad class of Frechet-Lie supergroups.

Abstract

For a broad class of Frechet-Lie supergroups we prove that there exists a correspondence between positive definite smooth superfunctions and matrix coefficients of unitary representations. We also give a characterization of linear functionals on the corresponding universal enveloping algebra which correspond to matrix coefficients of analytic unitary representations.