Abstract
Let A be a C∗-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (At)t∈T be a continuous field of operators in A such that the function t↦At is norm continuous on T and the function t↦‖At‖ is integrable. Then the following equality including Bouchner integrals holds(1)∫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.
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