Lectures on Amenability, Issue 1774
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 L^1(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.
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amenable Banach algebra analogue approximate identity approximation property Banach 21-bimodule Banach algebra 21 Banach modules Banach space biflat bimodule biprojective Banach algebra bounded approximate identity C(Sj C*-norm canonical Cb(G closed ideal cochain complex commutative completely positive Connes-amenable Corollary define Definition denote dual Banach algebra ea)a exact sequence Example Exercise finite finite-dimensional following are equivalent Fourier algebra functions g E G G is amenable group G Hence Hilbert space Hilbert space Sj Hochschild cohomology homological homomorphism ideal of 21 injective isometric isomorphism left approximate identity left Banach 21-module left Banach module left invariant mean Lemma let F Let G linear map linear spaces Ll(G locally compact group LUC(G Math operator amenable operator Banach algebra operator space Problem Proof Let proof of Theorem Proposition quasi-expectation right Banach right inverse semidiscrete Show strong operator topology subalgebra subspace suppose tensor product theory virtual diagonal VN(G von Neumann algebra