Existence of Approximately Macroscopically Unique States

Let H be an infinite dimensional separable Hilbert space and B(H) the \(C^*\)-algebra  of bounded operators on H. Suppose that \(T_1,T_2,..., T_n\) are self-adjoint operators in B(H). We show that, if commutators \([T_i, T_j]\) are sufficiently small in norm, then “Approximately Macroscopically Unique" states always exist for any values in a synthetic spectrum of the n-tuple of self-adjoint operators. This is achieved under the circumstance for which the n-tuple may not be approximated by commuting ones. This answers a question proposed by David Mumford for measurements in quantum theory. If commutators are not small in norm but small modulo compact operators, then “Approximate Macroscopic Uniqueness" states also exist.