A homological approach to the splitting theory of PLSwPLSw spaces

We give a homological approach to the splitting theory of PLSwPLSw spaces, that is strongly reduced projective limits of inductive limits of reflexive Banach spaces – a category that contains the PLS spaces that have been considered up to now. In particular we connect the problem under which conditions for given PLSwPLSw spaces E and X each short exact sequenceequation(⋆)0→X→Y→E→00→X→Y→E→0 of PLSwPLSw spaces splits to the vanishing of the Yoneda ExtPLSw1 functor in the category of PLSwPLSw spaces. Using the concept of exact categories this in turn is connected to the vanishing of the first derivative of the projective limit functor in a spectrum of operator spaces, thus generalizing results for special cases due to Bonet and Domański [2] and [3]. Furthermore, we apply the results to obtain a splitting theory for the space of Schwartz Distributions that includes the higher Ext functors, thus extending the result due to Domański and Vogt [13] respectively Wengenroth [40, (5.3.8)].