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Infinitely divisible states, cocycles and conditionally positive functionals on algebras. (Russian. English summary) Zbl 0876.47026

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

MSC:

47L50 Dual spaces of operator algebras
46K10 Representations of topological algebras with involution
22D15 Group algebras of locally compact groups
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