  We first present a generalization of the Robertson-Heisenberg uncertainty
principle. This generalization applies to mixed states and contains a
covariance term. For faithful states, we characterize when the uncertainty
inequality is an equality. We next present an uncertainty principle version for
real-valued observables. Sharp versions and conjugates of real-valued
observables are considered. The theory is illustrated with examples of
dichotomic observables. We close with a discussion of real-valued coarse
graining.
