Commutativity, Comonotonicity, and Choquet Integration of Self-adjoint Operators

In this work we propose a definition of comonotonicity for elements of B (H)sa, i.e., bounded self-adjoint operators defined over a complex Hilbert space H. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of B (H)sa that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over B (H)sa which are comonotonic additive, c- monotone, and normalized.