$$C^*$$-Quadratic Quantization

In the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual Heisenberg group introduced in Accardi et al. (Infin Dimens Anal Quantum Probab Relat Top 13:551–587, 2010) which makes the composition law much more transparent. In the second part of the paper the new coordinates are used to construct an inductive system of \(*\)-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode quadratic Weyl algebra. We prove that the inductive limit \(*\)-algebra is factorizable and has a natural localization given by a family of \(*\)-sub-algebras each of which is localized on a bounded Borel subset of \(\mathbb {R}\). Moreover, we prove that the family of quadratic analogues of the Fock states, defined on the inductive family of \(*\)-algebras, is projective hence it defines a unique state on the limit \(*\)-algebra. Finally we complete this \(*\)-algebra under the (minimal regular) \(C^*\)-norm thus obtaining a \(C^*\)-algebra.