Narrow Results

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.