 

Marc A. Rieffel
Standard deviation is a strongly Leibniz seminorm view print


Published: 
January 19, 2014

Keywords: 
standard deviation, Leibniz seminorm, C*algebra, matricial seminorm, conditional expectation 
Subject: 
Primary 46L53; Secondary 60B99 


Abstract
We show that standard deviation σ satisfies the Leibniz
inequality
σ(fg) ≦ σ(f)∥g∥ +
∥f∥σ(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 noncommutative
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.


Acknowledgements
The research reported here was supported in part by National Science Foundation grant DMS1066368


Author information
Department of Mathematics, University of California, Berkeley, CA 947203840
rieffel@math.berkeley.edu

