A homological approach to the splitting theory of PLSW spaces

作者：Dierolf Bernhard^*; Sieg Dennis

来源：Journal of Mathematical Analysis and Applications, 2016, 433(2): 1305-1328.

DOI：10.1016/j.jmaa.2015.08.030

• Splitting of short exact sequences
• Functors Ext(k)
• Exact categories
• Maximal exact structure
• Space of Schwartz distributions

We give a homological approach to the splitting theory of PLS, 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 PLSw spaces E and X each short exact sequence
0 -> X -> Y -> E -> 0
of PLSw spaces splits to the vanishing of the Yoneda Ext(PLSw)(1) fimctor in the category of PLSw 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 Domafiski [2,3]. 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 Domafiski and Vogt [13] respectively Wengenroth [40, (5.3.8)].

• 出版日期2016-1-15