Given a state on an algebra of bounded quantum mechanical observables, 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 terminology due to J. S. Bell). We also extend our results to the theory of algebras of unbounded observables (as developed by Kadison), and show how our results articulate a solid mathematical foundation for certain tenets of the orthodox Copenhagen interpretation of quantum theory.