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A Hilbert space operator representation of abelian po-groups of bilinear forms. (English) Zbl 1329.81097
Summary: The existence of a non-trivial singular positive bilinear form [B. Simon, J. Funct. Anal. 28, 377–385 (1978; Zbl 0413.47029)] yields that on an infinite-dimensional complex Hilbert space \(\mathcal H\) the set of bilinear forms \(\mathcal F(\mathcal H)\) is richer than the set of linear operators \(\mathcal V(\mathcal H)\). We show that there exists an structure preserving embedding of partially ordered groups from the abelian po-group \(\mathcal S_D(\mathcal H)\) of symmetric bilinear forms with a fixed domain \(D\) on a Hilbert space \(\mathcal H\) into the po-group of linear symmetric operators on a dense linear subspace of an infinite dimensional complex Hilbert space \(l_2(M)\). Moreover, if we restrict ourselves to the positive parts of the above mentioned po-groups, we can embed positive bilinear forms into corresponding positive linear operators.
MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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