Keller, Hans A.; Ochsenius A., Herminia Bounded operators on non-archimedean orthomodular spaces. (English) Zbl 0855.46049 Math. Slovaca 45, No. 4, 413-434 (1995). 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. Cited in 1 ReviewCited in 5 Documents 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 Keywords:valued fields; non-archimedean norm; projection theorem; orthogonal summand; infinite-dimensional orthomodular spaces; non-archimedean valuation; operator algebra of von Neumann type × Cite Format Result Cite Review PDF Full Text: EuDML References: [1] FÄSSLER-ULLMANN A.: Untersuchungen zu nichtklassischen Hilbertraumen. Ph.D. Thesis, Univ. of Zürich, 1982. [2] GROSS H., KÜNZI U. M.: On a class of orthomodular quadratic spaces. Enseign. Math. (2) 31 (1985), 187-212. · Zbl 0603.46030 [3] KELLER H. A.: Ein nicht-klassischer Hilbertscher Raum. Math. Z. 172 (1980), 41-49. · Zbl 0414.46018 · doi:10.1007/BF01182777 [4] KÜNZI U. M.: Nonclassical Hilbert spaces over valued fields. Master’s Thesis, Univ. of Zurich, 1980. [5] RIBENBOIM P.: Théorie des valuations. Les Presses de l’Université de Montreal, 1965. · Zbl 0139.26201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.