Extension of locally defined indefinite functions on ordered groups (Q1884740)

The authors aim to unify several previously known results concerning the extensions of some operator-valued functions. Their framework involves operators on Krein spaces \(({\mathcal K},\langle .,. \rangle)\) and domains in ordered abelian archimedean groups \((\Omega, +)\). The extension problem for a function \(f:[-2a,2a]\to L({\mathcal K})\) consists in the existence of a function \(F:\Omega\to L({\mathcal K})\) such that \(F|_{[-2a,2a]}=f\). The authors' key notion is that of \(k\)-indefinite function of archimedean type. A remarkable result (Theorem 7.5) shows that if \(\Gamma\) is a semi-archimedean group with the indefinite extension property, then \(\Gamma\times\mathbb{Z}^d\), endowed with the lexicographic order and the product topology, has the indefinite extension property. Consequently, \(\mathbb{Z}^n\) and \(\mathbb{R}\times\mathbb{Z}^n\) have the indefinite extension property.