An operator equality involving a continuous field of operators and its norm inequalities

Let AA be a C∗C∗-algebra, T be a locally compact Hausdorff space equipped with a probability measure P   and let (At)t∈T(At)t∈T be a continuous field of operators in AA such that the function t↦Att↦At is norm continuous on T   and the function t↦‖At‖t↦‖At‖ is integrable. Then the following equality including Bouchner integrals holdsequation(1)∫T|At-∫TAsdP|2dP=∫T|At|2dP-|∫TAtdP|2.∫TAt-∫TAsdP2dP=∫T|At|2dP-∫TAtdP2.This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten p-norm inequalities.