The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group (Q2781298)

Generalizing well-known results of M. G. Krein, Y. M. Berezansky, I. M. Gali in [\textit{J. Friedrich} and \textit{L. Klotz}, Rep. Math. Phys. Vol. 26, No. 1, 45-65 (1988; Zbl 0681.43008)] the authors proved that, given \(0 < a < \infty\) and a topological group \(G\), any strongly continuous positive definite function \(f: (-a,a) \times G \rightarrow \mathcal L(\mathcal H)\) admits a positive definite extension to \(\mathbb R \times G\). In the paper under review, the author generalizes the result by omitting the continuity requirement and by replacing \(\mathbb R\) by an ordered Abelian group (Theorem 2.1).