Interacting Fock space versus full Fock module; Volterra Preprint

Abstract. We present several examples where moments of creators and an-nihilators on an interacting Fock space may be realized as moments of creators and annihilators on a full Fock module. Motivated by this experience we an-swer the question, whether such a possibility exists for arbitrary interacting Fock spaces, in the affirmative sense. We treat the problem in full algebraic generality. As a by-product, we find a new notion of positivity for ∗–algebras which allows to construct tensor products of Hilbert modules over ∗–algebras. Finally, we consider a subcategory of interacting Fock spaces which are embeddable into a usual full Fock space. We see that a creator a∗(f) on the interacting Fock space is represented by an operator κ`∗(f), where `∗(f) is a usual creator on the full Fock space and κ is an operator which does not change the number of particles. In the picture of Hilbert modules the one-particle sector is replaced by a two-sided module over an algebra which contains κ. Therefore, κ may be absorbed into the creator, so that we are concerned with a usual creator. However, this creator does not act on a Fock space, but rather on a Fock module. 1