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).