CONTINUITY OF SYMPLECTICALLY ADJOINT MAPS AND THE ALGEBRAIC STRUCTUREOF HADAMARD VACUUM REPRESENTATIONS FOR QUANTUM-FIELDS ON CURVED SPACETIME

We derive for a pair of operators on a symplectic space which are adjo ints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar p roduct on the symplectic space dominating the symplectic form, then th ey are bounded with respect to a one-parametric family of scalar produ cts canonically associated with the initially given one, among them be ing its ''purification''. As a typical example we consider a scalar fi eld on a globally hyperbolic spacetime governed by the Klein-Gordon eq uation; the classical system is described by a symplectic space and th e temporal evolution by symplectomorphisms (which are symprectically a djoint to their inverses). A natural scalar product is that inducing t he classical energy norm, and an application of the above result yield s that its ''purification'' induces on the one-particle space of the q uantized system a topology which coincides with that given by the two- point functions of quasifree Hadamard states. These findings will be s hown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-dualit y (and also split-and type III1-properties). A brief review of this ci rcle of notions, as well as of properties of Hadamard states, forms pa rt of the article.