Normal forms on partial O*-algebras

1089-7658

AIP Digital Archive

Mathematics

Physics

In the algebraic formulation of quantum theories, a state is often represented by a normal linear functional on some *-algebra @FA of operators on a Hilbert space, i.e., a functional of the form f(X)=Tr TX for some trace class operator X. The question is whether every strongly positive (i.e., positive on positive elements) linear functional is normal. Criteria for this statement to be true are well known in the case where @FA is a *-algebra of bounded operators (C*- or W*-algebra) or a *-algebra of unbounded operators (O*-algebra). Those results are extended to the case where @FA is a (weak) partial *-algebra of closable operators, both for linear functionals and for sesquilinear forms.

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