  Given a state on an algebra of bounded quantum-mechanical observables (the
self-adjoint part of a C*-algebra), we investigate those subalgebras that are
maximal with respect to the property that the given state's restriction to the
subalgebra is a mixture of dispersion-free states---what we call maximal
"beable" subalgebras (borrowing a terminology due to J. S. Bell). We also
extend our investigation to the theory of algebras of unbounded observables (as
developed by R. Kadison), and show how our results articulate a solid
mathematical foundation for central tenets of the orthodox Copenhagen
interpretation of quantum theory (such as the joint indeterminacy of
canonically conjugate observables, and Bohr's defense of the completeness of
quantum theory against the argument of Einstein, Podolsky, and Rosen).
