A pseudo-Euclidean representation of conditionally positive mappings (Q1177826)

Motivated by the Itô stochastic calculus the author investigates semigroups of states on \(*\)-semirings. (The \(*\)-semiring is defined as an associative semigroup with an involution, an identity and a partially defined sum operation.) Let \(B\) be a \(*\)-semiring and \((\rho_ t)\) be a continuous collection of additive functions defined on \(B\) with values in the set \({\mathcal L} (D_ 1, D_ 2)\) of linear operators between pre- Hilbert spaces \(D_ 1\) and \(D_ 2\) \((D_ 1 \subset D_ 2)\). Under some natural assumptions it is proved that the derivative \(\lambda\) of \((\rho_ t)\) at 0 is an additive and conditionally positive function on \(B\). Moreover, the mapping \(\lambda\) is described by means of a representation of \(B\) on \(K\) and an additive mapping of \(B\) into \({\mathcal L} (D_ 1, K)\), where \(K\) is a tensor product of a Euclidean space and some Hilbert space and \({\mathcal L} (D_ 1, K)\) is the set of weakly continuous operators, respectively. The equivalent realisation of \(\lambda\) of a pseudo-Hilbert space is investigated.