From discrete to continuous monotone $C^*$-algebras via quantum central limit theorems

TL;DR

This paper demonstrates how joint distributions of creation and annihilation operators in monotone Fock spaces can be approximated by quantum central limit theorems within $C^*$-algebras, extending to processes via an invariance principle.

Contribution

It introduces a method to realize joint distributions as quantum central limits of operators in discrete monotone $C^*$-algebras, generalizing to processes.

Findings

Joint distributions approximated by quantum central limit theorems.

Construction applies to processes through an invariance principle.

Extension from discrete to continuous monotone $C^*$-algebras.

Abstract

We prove that all finite joint distributions of creation and annihilation operators in Monotone and anti-Monotone Fock spaces can be realized as Quantum Central Limit of certain operators on a $C^*$-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone $C^*$-algebras, the weight being the above cited test functions. The construction is then generalized to processes by an invariance principle.