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Angle Bisection and Orthoautomorphisms in Hilbert Lattices

Published online by Cambridge University Press:  20 November 2018

Ronald P. Morash
Ronald P. Morash*
Affiliation:
University of Massachusetts, Amherst, Massachusetts; University of Michigan, Dearborn, Michigan
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The lattices of all closed subspaces of separable, infinitedimensional Hilbert space (real, complex, and quaternionic) share the following purely lattice-theoretic properties. Each is complete, orthocomplemented, atomistic, irreducible, separable, M-symmetric, and orthomodular [2]. We will call any lattice possessing these seven properties a Hilbert lattice. The general situation which motivates the investigations of this paper concerns infinite-dimensional Hilbert lattices (the dimension of a Hilbert lattice being the cardinality of any maximal family of orthogonal atoms). There are several lattice theoretic properties, possessed by the three canonical lattices, whose only known proofs involve the analytic properties of the underlying Hilbert space, that is, there is no known purely lattice-theoretic proof of these properties.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 2 , April 1973 , pp. 261 - 272
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Maeda, F. and Maeda, S., Theory of symmetric lattices (Springer-Verlag, Berlin, 1970).Google Scholar
2. Morash, R. P., The orthomodular identity and metric completeness of the coordinatizing division ring, Proc. Amer. Math. Soc. 27 (1971), 446–448.Google Scholar
3. Morash, R. P., supplement to [2], Proc. Amer. Math. Soc. 29 (1971), 627.Google Scholar
4. Morash, R. P., Orthomodularity and the direct sum of division subrings of the quaternions, Proc. Amer. Math. Soc. 36 (1972), 63–68.Google Scholar