Angle Bisection and Orthoautomorphisms in Hilbert Lattices

Abstract

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.