Infinitely divisible states, 1-cocycles, and conditionally positive functionals on algebras

Pairs of algebras in duality whose state spaces are semigroups (the duality in the sense of Vershik) are considered. A class of such algebras, which are called equipped, is introduced. For these algebras, the cone of conditionally positive functionals and its relationship with the geometry of the dual object and the 1-cocycles on with values in the *-representations are studied. In the case of group algebras, a symmetric construction describing infinitely divisible measures and states is given. Bibliography: 10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 146–162. Translated by B. M. Bekker.