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

Let image be a C*-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (At)tset membership, variantT be a continuous field of operators in image such that the function tmaps toAt is norm continuous on T and the function tmaps todouble vertical barAtdouble vertical bar is integrable. Then the following equality including Bouchner integrals holds ∫TAt-∫TAsdP2dP=∫TAt2dP-∫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.