$C^*$-non-linear second quantization

Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators are indexed by elements of an appropriate test function space (second quantization), can be done at Lie algebra level. A way to bypass the difficulties of constructing a (non trivial) Hilbert space representation is to try and construct directly a $C^*$-algebra rep- resentation and then to look for its Hilbert space representations. In usual (linear) quantization, this corresponds to the construction of the Weyl $C^*$-algebra. In this paper, we produce such a construction for the above mentioned polynomial extension of the Weyl $C^*$-algebra. The result of this construction is a factorizable system of local alge- bras localized on bounded Borel subsets of $\mathbb{R}$ and obtained as induc- tive limit of tensor products of finite sets of copies of the one mode $C^*$-algebra. The $C^*$-embeddings of the inductive system require some non{trivial re{scaling of the generators of the algebras involved. These re{scalings are responsible of a $C^*$-analogue of the "no-go" theorems, first met at the level of Fock second quantization, namely the proof that the family of Fock states defined on the inductive family of $C^*$-algebras is projective only in the linear case (i.e. the case of the usual Weyl algebra). Thus the solution of the representa- tion problem at $C^*$-level does not automatically imply its solution at Hilbert space level.