An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities
نویسندگان
چکیده
Let A be a C *-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (A t) t∈T be a continuous field of operators in A such that the function t → A t is norm continuous on T and the function t → A t is integrable. Then the following equality including Bouchner integrals holds T A t − T A s dP 2 dP = T |A t | 2 dP − T A t dP 2. (0.1) 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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