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On a class of orthomodular quadratic spaces. (English) Zbl 0603.46030

Let E be a K-vector space equipped with an anisotropic sesquilinear form and let \(\perp\) be the associated orthogonality relation. E is called orthomodular (Kaplansky) (or hilbertian (Varadarajan)) iff E is infinite dimensional and all \(\perp\)-closed linear subspaces of E are splitting, \(X=(X^{\perp})^{\perp}\Rightarrow E=X\oplus X^{\perp}\). In 1979 Hans A. Keller discovered the first example of an orthomodular space different from classical Hilbert space (over \(K={\mathbb{R}},{\mathbb{C}},{\mathbb{H}})\). It is over a non-archimedean ordered field K; H. Gross noticed that it is possible and advantageous to view K as the field of formal power series \({\mathbb{Q}}((\Gamma))\) with the ordered abelian group \(\Gamma ={\mathbb{Z}}^{({\mathbb{N}})}\). By throwing out orderings in favour of valuations the original ideas of Keller became much more perspicuous and the construction of new examples at will became possible. The paper under review is an introduction into this topic. As a side result a list of long standing questions could be answered [H. Gross, Contributions to general algebra 3, Proc. Conf., Vienna 1984, 181-190 (1985; Zbl 0573.06008); U.-M. Künzi: A Hilbert lattice with a small automorphism group, to appear in Can. J. Math.]. The methods of inifnite dimensional quadratic form theory developped here have also made possible the construction of ”propositional systems” that admit different negations [H. Gross, Different orthomodular orthocomplementations on a vector subspace lattice, to appear in ORDER].

MSC:

46C99 Inner product spaces and their generalizations, Hilbert spaces
11E04 Quadratic forms over general fields
06C15 Complemented lattices, orthocomplemented lattices and posets
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
15A63 Quadratic and bilinear forms, inner products

Citations:

Zbl 0573.06008
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