Flat locally convex modules beyond the metrizable case

Abstract: We begin by surveying classical results (due to Johnson, Helemskii, and Sheinberg) on amenable Banach algebras and flat Banach modules. In particular, we discuss 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 look at some possible extensions of these concepts to more general locally convex 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, 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).

category theoryfunctional analysisrepresentation theory

Audience: researchers in the topic

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Additive categories between algebra and functional analysis

Series comments: Aims & Scope: Exchange ideas and foster collaboration between researchers from representation theory and functional analysis working on categorical aspects of the theory. In addition to research talks, there will be four mini-courses of introductory character.

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