XXXVII Workshop on Geometric Methods in Physics | 1-7.07.2018 |

VII School on Geometry and Physics | 25-29.06.2018 |

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## Alexander Helemskii## Amenability, Flatness and Measure AlgebrasOne of most known applications of the homological algebra to functional analysis is the positive answer to the following old conjecture: is it true that for an amenable measure algebra M(G) the respective group G must be discrete? We shall formulate such a theorem, explain its ingredients, discuss it, compare it with some other results, and give its detailed proof. We hope that the argument, being an interplay of various ideas and methods of algebra and analysis, will be interesting and instructive to young researchers. It involves the basic homological algebra in its functional-analytic version (notably the notion of a flat Banach module), classical functional analysis (in particular, properties of the weak∗ topology in the second dual of a Banach space) and, finally, the presence of some exotic measurable subsets in a given non-discrete locally compact group. |

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University of Bialystok |

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