A Lévy-Khinchine formula for semigroups and related problems: An adapted spaces approach (Q1916780)

Let \(X\) be a locally compact Hausdorff space. A linear space \(V\) of real continuous functions on \(X\) is called adapted if (i) \(V=V_+ -V_+\), where \(V_+ = \{f\in V: f\geq 0\}\) and (ii) for every \(x\in X\) there is an \(f\in V_+\) such that \(f(x)\geq 0\). Using a theorem proved in Section 1 on adapted spaces, the author obtains an integral representation for a set of negative definite functions defined as a commutative semigroup with neutral element. This is done in Section 2. It is proved that a completely monotonic (resp. alternating) function, defined on a commutative semigroup (with or without neutral element) is completely positive (resp. negative) definite. A characterisation for completely monotonic (resp. alternating) functions defined on \(N^*\) is also provided. Finally, the author considers a Stieltjes type problem for functions defined on \(N^*\).