\b{eta}-KMS Green functions generating functionals - the convexity based approach

The notion of the stochastically positive b{eta}-KMS (Euclidean time ) Green functionals on ( the abelian sectors of ) the CCR-algebras in Weyl form has been introduced .The main observation is that the essential properties of such functionals are stable against taking convex superpositions of them. Starting from the free Bose matter describing functionals( the condensed state situation is included as well) a constructive approach to the constructions of a new ,of such type functionals is presented. In particular , starting from the free Bose matter Green functionals a class of not quasi-free thermal functionals, together with the corresponding modular structures have been constructed. Some elementary properties of models constructed are being presented.