In previous works the author introduced the notions of \({\mathcal Z}\)-algebras, \({\mathcal Z}\)-quantum probability space, \({\mathcal Z}\)-quantum random variables, and Bose \({\mathcal Z}\)-independence [cf. Commun. Math. Phys. 192, No. 3, 569-604 (1998; Zbl 0928.46063)], where \({\mathcal Z}\) is some fixed unital \({C^*}\)-algebra. His approach follows the idea of operator-valued random variables of D. Voiculescu [Recent advances in operator algebras, Astérisque 232, 243-275 (1995; Zbl 0839.46060)], but to define tensor products of \({\mathcal Z}\)-algebras, Bose \({\mathcal Z}\)-independence, symmetric Fock modules, etc., it is necessary to restrict to a subclass of \({\mathcal Z}\)-\({\mathcal Z}\)-modules called centered by the author. Key examples of families of Bose \({\mathcal Z}\)-independent \({\mathcal Z}\)-random variables are creation and annihilation operators on symmetric Fock modules. In this paper the author proves a central limit theorem and shows that the central limit distributions can be represented by an algebra of creators and annihilators on a symmetric Fock module.