Lectures on Amenability

The notion of amenability has its origins in the beginnings of modern measure theory: Does a finitely additive set function exist which is invariant under a certain group action? Since the 1940s, amenability has become an important concept in abstract harmonic analysis (or rather, more generally, in the theory of semitopological semigroups). In 1972, B.E. Johnson showed that the amenability of a locally compact group G can be characterized in terms of the Hochschild cohomology of its group algebra Lsub1(G): this initiated the theory of amenable Banach algebras. Since then, amenability has penetrated other branches of mathematics, such as von Neumann algebras, operator spaces, and even differential geometry. Lectures on Amenability introduces second year graduate students to this fascinating area of modern mathematics and leads them to a level from where they can go on to read original papers on the subject. Numerous exercises are interspersed in the text. 0. Paradoxical decompositions....Pages 1-15 1. Amenable, locally compact groups....Pages 17-36 2. Amenable Banach algebras....Pages 37-61 3. Examples of amenable Banach algebras....Pages 63-81 4. Amenability-like properties....Pages 83-117 5. Banach homology....Pages 119-139 6. C * - and W * -algebras....Pages 141-190 7. Operator amenability....Pages 191-207 8. Geometry of spaces of homomorphisms....Pages 209-219 Open problems....Pages 221-229 A Abstract harmonic analysis....Pages 231-241 B Tensor products....Pages 243-254 C Banach space properties....Pages 255-263 D Operator spaces....Pages 265-274 List of Symbols....Pages 275-280 References....Pages 281-288 Index....Pages 289-296