Extension of local positive definite $\mathbb{Z}_{2}^{n}$-superfunctions

TL;DR

This paper extends local positive definite $Z_2^n$-superfunctions to global ones on $Z_2^n$-Lie supergroups by introducing a boundedness concept, ensuring the extension's positivity.

Contribution

It introduces a boundedness condition for $Z_2^n$-superfunctions and proves that bounded local positive definite functions extend globally on $Z_2^n$-Lie supergroups.

Findings

Local positive definite $Z_2^n$-superfunctions can be extended globally under boundedness.

The extension preserves positive definiteness.

The work generalizes classical extension results to the supergroup setting.

Abstract

By defining a concept of boundedness for $ \mathbb{Z}_2^n $-superfunction, we show that each local positive definite $ \mathbb{Z}_2^n $-superfunctions, which is bounded in some sense, on a $ \mathbb{Z}_2^n $-Lie supergroup has a positive definite extension to all of the $\mathbb{Z}_{2}^{n}$-Lie supergroup.