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Bounded operators on non-archimedean orthomodular spaces. (English) Zbl 0855.46049

Summary: A single most important fact underlying the theory of infinite-dimensional Hilbert spaces is embodied in the projection theorem: every orthogonally closed subspace is an orthogonal summand. A hermitian space which enjoys this property is said to be orthomodular. Besides the obvious Hilbert spaces, there do exist other infinite-dimensional orthomodular spaces, examples of which have so far only been constructed over complete fields with a non-archimedean valuation. In this article, we study bounded linear operators on such spaces, many features of which are found to diverge sharply from those of bounded operators in the classical Hilbert space setting. In particular, we construct an operator algebra of von Neumann type that contains no orthogonal projections at all. For this algebra a representation theorem is derived, which implies that it is commutative.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
12J25 Non-Archimedean valued fields
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References:

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