Standard deviation is a strongly Leibniz seminorm

Abstract: We show that standard deviation $\s$ satisfies the Leibniz inequality $\s(fg) \leq \s(f)|g| + |f|\s(g)$ for bounded functions f, g on a probability space, where the norm is the supremum norm. A related inequality that we refer to as "strong" is also shown to hold. We show that these in fact hold also for non-commutative probability spaces. We extend this to the case of matricial seminorms on a unital C*-algebra, which leads us to treat also the case of a conditional expectation from a unital C*-algebra onto a unital C*-subalgebra.