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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...
We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C*- and W*-algebras; we show that interpolation space techniques can be used in place of GNS type arguments....
Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map (respectively, ) acting on the operators of the surjective (respectively, injective) hull of such that (respectively, ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving and ....
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