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We present a user-friendly version of a double operator integration theory which still
retains a capacity for many useful applications. Using recent results from the latter
theory applied in noncommutative geometry, we derive applications to analogues of the
classical Heinz inequality, a simplified proof of a famous inequality of
Birman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods are
sufficiently strong to treat these...
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