On Inner Characterizations of Pseudo-Krein and Pre-Krein Spaces

Abstract

For developing a theory of ${}^{\ast }$-representations of ${}^{\ast }$-algebras $\mathcal{A}$ on (indefinite) inner product spaces $E,(\cdot ,\cdot )$, the classes of pseudo-Krein and pre-Krein spaces which admit a Hilbert space structure and exactly one maximal Hilbert space structure, respectively, are introduced. Inner characterizations for the topological structure of $E,(\cdot ,\cdot )$ to be a pseudo-Krein and pre-Krein space, respectively, are given. The results are illustrated by some examples.