Lectures on Amenability

ISBN-13: 9783540428527 / Angielski / Miękka / 2001 / 302 str.

V. Runde; Volker Runde

Opis

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 DEGREES1(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.

Spis treści

0 Paradoxical decompositions
0.1 The Banach-Tarski paradox
0.2 Tarski's theorem
0.3 Notes and comments

1 Amenable, locally comact groups
1.1 Invariant means on locally compact groups
1.2 Hereditary properties
1.3 Day's fixed point theorem
1.4 Representations on Hilbert space
1.5 Notes and comments

2 Amenable Banach algebras
2.1 Johnson's theorem
2.2 Virtual and approximate diagonals
2.3 Hereditary properties
2.4 Hochschild cohomology
2.5 Notes and comments

3 Exemples of amenable Banach algebras
3.1 Banach algebras of compact operators
3.2 A commutative, radical, amenable Banach algebra
3.3 Notes and comments

4 Amenability-like properties
4.1 Super-amenability
4.2 Weak amenability
4.3 Biprojectivity and biflatness
4.4 Connes-amenability
4.5 Notes and comments

5 Banach homology
5.1 Projectivity
5.2 Resolutions and Ext-groups
5.4 Flatness and injectivity
5.4 Notes and Comments

6 C* and W*-algebras
6.1 Amenable W*-algebras
6.2 Injective W*-algebras
6.3 Tensor products of C*- and W*-algebras
6.4 Semidiscrete W*-algebras
6.5 Normal, virtual diagonals
6.6 Notes and comments

7.1 Bounded approximate identities for Fourier algebras
7.2 (Non-)amenability of Fourier
7.3 Operator amenable operator Banach algebras
7.4 Operator amenability of Fourier algebras
7.5 Operator amenability of C*-algebras
7.6 Notes and comments

8 Geometry of spaces of homomorphisms
8.1 Infinite-dimensional differential geometry
8.2 Spaces of homomorphisms
8.3 Notes and Comments

Open problems

A Abstract harmonic analysis
A.1 Convolution of measures and functions
A.2 Invariant subspaces of L(infinity symbol)(G)
A.3 Regular representations on Lp(G)
A.4 Notes and comments

B.1 The algebraic tensor products
B.2 Banach space tensor products
B.2.1 The injective tensor product
B.2.2 The projective tensor product
B.3 The Hilbert space tensor product
B.4 Notes and comments

C Banach space properties
C.1 Approximation properties
C.2 The Radon-Nikokym property
C.3 Local theory of Banach spaces
C.4 Notes and comments

D Operator spaces
D.1 Abstract and concrete operator spaces
D.2 Completely bounded maps
D.3 Tensor products of operator spaces
D.4 Operator Banach algebras
D.5 Notes and comments
List of symbols
References
Index