Seminar "Functional analysis and noncommutative geometry". Speaker: Alexei Pirkovskii

We begin by surveying classical results (mostly due to Johnson, Helemskii, and Sheinberg) on amenable Banach algebras and flat Banach modules. In particular, we prove Helemskii-Sheinberg's theorem which states that a Banach algebra A is Johnson amenable if and only if its unitization is a flat Banach A-bimodule.
Next we discuss some possible extensions of these concepts to more general topological algebras and modules. The "naive" generalization of the notion of a flat Banach module to the nonmetrizable setting turns out to be not very useful. We suggest a modified definition, and we show how it works in concrete situations.
As an application (if time permits), we give a characterization of amenable co-echelon algebras obtained in our recent perprint with Krzysztof Piszczek. Curiously, the nonmetrizable case requires some essentially new tools (as compared to the Banach case), not only from analysis, but also from homological algebra (t-structures and their hearts).

This talk was scheduled for November 26, but was cancelled for technical reasons.