The Gelfand-Naimark-Segal construction for unitary representatins of $\mathbb Z_2^n$-graded Lie supergroups

TL;DR

This paper develops a Gelfand-Naimark-Segal construction that creates a correspondence between cyclic unitary representations and positive definite superfunctions for a broad class of $ ext{Z}_2^n$-graded Lie supergroups, advancing the mathematical framework of supergroup representations.

Contribution

It extends the Gelfand-Naimark-Segal construction to $ ext{Z}_2^n$-graded Lie supergroups, providing a new method to analyze their unitary representations.

Findings

Established a GNS construction for $ ext{Z}_2^n$-graded Lie supergroups.

Connected cyclic unitary representations with positive definite superfunctions.

Generalized the framework for supergroup representation theory.

Abstract

We establish a Gelfand-Naimark-Segal construction which yields a correspondence between cyclic unitary representations and positive definite superfunctions of a general class of $\mathbb Z_2^n$-graded Lie supergroups.