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

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