Abstract
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 quadraticWeyl 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 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.