Bernstein theorems and transformations of correlation measures in statistical physics

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products $(\,\cdot\,{,}\,\star)$ and the maps $K$, $K^{-1}$ which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): $F(\gamma)\to \widehat F_\cup(\gamma)=\{F(\alpha\cup\beta)\}_{\alpha,\beta\subset\gamma}$. For the operators $\widehat F_\cup(\gamma)$ and $\widehat F_\cap(\gamma)$, we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function $F(\gamma)$.