Abstract
A relatively orthocomplemented lattice L is a lattice in which every interval is an orthocomplemented sublattice. An orthogonally scattered measure ξ on L is a Hilbert space valued abstract measure over L such that ξ( e ) ⊥ ξ( f ) whenever e ⊥ f in L . The properties of so generalized c.a.o.s. measures are studied, the representation theorem has been proved: every H -valued c.a.o.s. measure ξ on L is of the form ξ(e) = Φ(e)x , where x ε H , and Φ is a lattice orthohomomorphism from L into Proj ( H ). The results generalize those in [21]. Their suitability for many applications has been demonstrated, including duality theory for some inductive-projective limits of Hilbert spaces and quantum probability.
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