Infinitelydivisible states, $1$-cocycles and conditionally positive functionals on algebras

The duality pairs of algebras are considered. It means that spaces of states are semigroups (duality in Vershik sense). We introduce a class algebras, named equipped, and investigate the cone $CL(\mathfrak A)$ of conditionally positive functionals on algebra $\mathfrak A$, connections between $CL(\mathfrak A)$, geometry of dual object $\mathfrak A$ and $1$-cocycles on $\mathfrak A$ to $*$-representations. In group algebras case we have a symmetric construction to describe infinitelydivisible measures and states. Bibliography: 10 titles.