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Algebras of bounded operators on nonclassical orthomodular spaces. (English) Zbl 0809.46094

Summary: A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite- dimensional examples constructed over certain non-Archimedean valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebra \(\mathcal A\) of von Neumann type that contains no orthogonal projections at all. For operators in \(\mathcal A\) we establish a representation theorem from which we deduce that \(\mathcal A\) is commutative. We then focus on a subalgebra \(\mathcal H\) which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of \(\mathcal H\) are given.

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)
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References:

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