On GNS representations on inner product spaces - I. The structure of the representation space

A generalization of the GNS construction to hermitian linear functionals W defined on a unital *-algebra A is considered. Along these lines, a continuity condition (H) upon W is introduced such that (H) proves to be necessary and sufficient for the existence of a J-representation x --> pi(W)(x), z is an element of A, on a Krein space H. The property whether or not the Gram operator J leaves the (common and invariant) domain D of the representation invariant is characterized as well by properties of the functional W as by those of D. Furthermore, the interesting class of positively dominated functionals is introduced and investigated. Some applications to tensor algebras are finally discussed.