We write S∞ to denote the unit sphere of `, the set of all z in ` for which 〈z, z〉 = 1 (this set is often called the complex Hilbert sphere). A continuous complex-valued function f defined on the closed unit disc D := {ζ ∈ C : |ζ| ≤ 1} is said to be positive definite of order n on S∞ if, for any set of n distinct points z1, . . . , zn on S∞, the n×n matrix A with ij-entry given by f(〈zi, zj〉) i...

in the present paper, among other results, a decomposition formula is given for the w-bounded continuous negative definite functions of a topological *-semigroup s with a weight function w into a proper h*-algebra a in terms of w-bounded continuous positive definite a-valued functions on s. a generalization of a well-known result of k. harzallah is obtained. an earlier conjecture of the author ...