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

We prove that all finite joint distributions of creation and annihilation operators in monotone and anti-monotone Fock spaces can be realised as Quantum Central Limit of certain operators in 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 weights being related to the above cited test functions.