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17 December, 2010

With term over (bar an exam to invigilate and help mark) I have been trying to get back to some unfinished bits of research. It’s been a bit harder than I expected to pick up loose ends where they were left and remember how they were supposed to be tied off or plaited.

Anyway: on the grounds that people who follow fashionable mathematics are supposed to blog about as it happens to lend some razzmatazz or immediacy, I thought I’d do the same for something rather less trendy but (to some of us) just as interesting. For, a couple of weeks ago, in the modestly titled 3-page preprint

arxiv 1012.1488 U. Bader, T. Gelander, N. Monod, A fixed point theorem for L^1 spaces

the authors not only establish a fixed point theorem as stated in their title, but use it to give what amounts to a one page solution to the so-called derivation problem for group algebras. (Strictly speaking, this assume some preliminary reductions of the derivation problem to statements about group 1-cocycles, but these reductions are fairly standard and have long been known to specialists.)

This is extremely startling — to me at least — because the derivation problem, open since the 1960s and which resisted the efforts of several researchers, was only resolved this decade in a formidable paper of V. Losert in the Annals of Mathematics. Not only is the argument of Bader, Gelander and Monod shorter, but it uses less technical machinery and puts the result in a more general context.

My plan is to do a series of posts – probably starting in the new year – on the derivation problem and the new solution, as a way of working through the details myself. Matthew Daws has recently written up some short notes on the new proof, which you can find on his website; I will probably take a slower and more pedestrian approach, with more digressions on some of the background concerning derivations from Banach algebras.

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