ON GNS REPRESENTATIONS ON INNER-PRODUCT SPACES - I - THE STRUCTURE OFTHE REPRESENTATION SPACE

A generalization of the GNS construction to hermitian linear functiona ls 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 propert y 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 inte resting class of positively dominated functionals is introduced and in vestigated. Some applications to tensor algebras are finally discussed .