Interloper/dilettante. “Functional analyst”, who sometimes wonders if the term meant something different in Britain that it does in North America. Keeps finding category-theoretic things all over the place in old analysis, never mind the brave new world of NCG, and is starting to feel that they should be collated somewhere.
Some things I am considering trying to write about, or contribute to. (This is mainly meant as a list to remind or nag me to write things down.) * projective, injective and flat Banach spaces (both in the metric and topological categories of Banach spaces) * RKHS * the bidualization monad on Banach spaces * Kaijser and Wick-Pelletier’s work on interpolation * Arens-Eells spaces, a.k.a. the adjunction between Banach spaces and metric spaces. * Richard Arens’s paper using “phyla”, or: where Banach algebras and symmetric monoidal categories should have met. (There is now some stuff on Arens products? on the Banach algebras page.) * Categori(c)al basics of operator spaces: MAX, MIN, and various tensor products * functorial aspects of the group von Neumann algebra construction * cabbages * kings * whether pigs have wings
J.M.F. Castillo. The hitchhiker guide to categorical Banach space theory. Part I. Extracta Math. 25(2) (2010), 103–149.
(Currently freely available on the journal’s website )