Decomposition of conditionally positive definite functions on commutative hypergroups (Q431164)

The author defines so-called conditionally positive definite functions on a commutative hypergroup \(K\) w.r.t. a finite set \(U\) of characters. For \(U=\{\mathbf 1\}\), this concept agrees with the well-known of conditionally positive definite (or, up to a sign, negative definite) functions. The author presents some canonical examples and proves a representation theorem for bounded conditionally positive definite functions under the assumption that the ideal in \(L^1(K)\) associated with \(U\) admits a bounded approximate identity. For the classical case \(U=\{\mathbf 1\}\), this existence is known by a result of Wolfenstetter, and the representation result there was already derived by the reviewer in [\textit{M. Voigt}, Lect. Notes Math. 1379, 376--388 (1989; Zbl 0672.60016)]. Other cases were studied later on by Filbir. The author here mainly uses this approach. Unfortunately, no comments are given, for which \(U\) with \(|U|\geq2\) his condition on bounded approximate identities is satisfied.