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We derive for a pair of operators on a symplectic space which are adjoints ofeach other with respect to the symplectic form (that is, they are sympleticallyadjoint) that, if they are bounded for some scalar product on the symplecticspace dominating the symplectic form, then they are bounded with respect to aone-parametric family of scalar products canonically associated with theinitially given one, among them being its ``purification''. As a typicalexample we consider a scalar field on a globally hyperbolic spacetime governedby the Klein-Gordon equation; the classical system is described by a symplecticspace and the temporal evolution by symplectomorphisms (which aresymplectically adjoint to their inverses). A natural scalar product is thatinducing the classical energy norm, and an application of the above resultyields that its ``purification'' induces on the one-particle space of thequantized system a topology which coincides with that given by the two-pointfunctions of quasifree Hadamard states. These findings will be shown to lead tonew results concerning the structure of the local (von Neumann)observable-algebras in representations of quasifree Hadamard states of theKlein-Gordon field in an arbitrary globally hyperbolic spacetime, such as localdefiniteness, local primarity and Haag-duality (and also split- and typeIII_1-properties). A brief review of this circle of notions, as well as ofproperties of Hadamard states, forms part of the article.Comment: 42 pages, LaTeX. The Def. 3.3 was incomplete and this has been  corrected. Several misprints have been removed. All results and proofs remain  unchange

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime