We present a user-friendly version of a double operator integration theory which stillretains a capacity for many useful applications. Using recent results from the lattertheory applied in noncommutative geometry, we derive applications to analogues of theclassical Heinz inequality, a simplified proof of a famous inequality ofBirman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods aresufficiently strong to treat these inequalities in the setting of symmetric operator normsin general semifinite von Neumann algebras.
