Geodesic
Infinitelydivisible states, $1$-cocycles and conditionally positive functionals on algebras
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 14, Tome 215 (1994), pp. 146-162 
S. I. Karpushev. Infinitelydivisible states, $1$-cocycles and conditionally positive functionals on algebras. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 14, Tome 215 (1994), pp. 146-162. http://geodesic.mathdoc.fr/item/ZNSL_1994_215_a9/
@article{ZNSL_1994_215_a9,
     author = {S. I. Karpushev},
     title = {Infinitelydivisible states, $1$-cocycles and conditionally positive functionals on algebras},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {146--162},
     year = {1994},
     volume = {215},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1994_215_a9/}
}
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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.