Lectures on amenability.

*(English)*Zbl 0999.46022
Lecture Notes in Mathematics. 1774. Berlin: Springer. xiii, 296 p. EUR 43.95/net; sFr. 73.00; £31.00; $ 59.80 (2002).

The theory of amenable Banach algebras was initiated by B. E. Johnson in 1972 when he characterized amenability of a locally compact group \(G\) in terms of the Hochschild cohomology of its group algebra \(L^1(G)\). Since then, amenable Banach algebras and amenability in general have been an active field of research.

The declared aim of these notes is to introduce second year graduate students to amenable Banach algebras and related questions and leads them to a level from where they can go on to read original research papers on the subject. My opinion is that the author achieved his goals and I recommend this very nicely written book to the interested reader.

The following is a brief description of the contents of the Lecture Notes. The book starts with paradoxical decompositions, moves through amenable, locally compact groups to amenable Banach algebras, and treats amenability for \(C^*\)- and von Neumann algebras in great detail. Recent results and developments are also covered. For instance, C. J. Read’s example of a commutative radical Banach algebra and Z.-J. Ruan’s notion of operator amenability are discussed.

There are numerous exercises, varying greatly in their degree of difficulty, in the text as well as notes and comments sections at the end of each chapter.

The following are the titles of all chapters of the book : 0. Paradoxical decompositions 1. Amenable, locally compact groups 2. Amenable Banach algebras 3. Examples of amenable Banach algebras 4. Amenability-like properties 5. Banach homology 6. \(C^*\)- and \(W^*\)-algebras 7. Operator amenability 8. Geometry of spaces of homomorphisms.

The book ends with a list of open problems and with four appendices about abstract harmonic analysis, tensor products, Banach space properties and operator spaces. There are some missprints or small errors - for an errata and for updates, the author maintains a Web page at http://www.math.ualberta.ca/~runde/files/errata.pdf.

The declared aim of these notes is to introduce second year graduate students to amenable Banach algebras and related questions and leads them to a level from where they can go on to read original research papers on the subject. My opinion is that the author achieved his goals and I recommend this very nicely written book to the interested reader.

The following is a brief description of the contents of the Lecture Notes. The book starts with paradoxical decompositions, moves through amenable, locally compact groups to amenable Banach algebras, and treats amenability for \(C^*\)- and von Neumann algebras in great detail. Recent results and developments are also covered. For instance, C. J. Read’s example of a commutative radical Banach algebra and Z.-J. Ruan’s notion of operator amenability are discussed.

There are numerous exercises, varying greatly in their degree of difficulty, in the text as well as notes and comments sections at the end of each chapter.

The following are the titles of all chapters of the book : 0. Paradoxical decompositions 1. Amenable, locally compact groups 2. Amenable Banach algebras 3. Examples of amenable Banach algebras 4. Amenability-like properties 5. Banach homology 6. \(C^*\)- and \(W^*\)-algebras 7. Operator amenability 8. Geometry of spaces of homomorphisms.

The book ends with a list of open problems and with four appendices about abstract harmonic analysis, tensor products, Banach space properties and operator spaces. There are some missprints or small errors - for an errata and for updates, the author maintains a Web page at http://www.math.ualberta.ca/~runde/files/errata.pdf.

Reviewer: Catalin Badea (Villeneuve d’Ascq)

##### MSC:

46H20 | Structure, classification of topological algebras |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

43A07 | Means on groups, semigroups, etc.; amenable groups |

46L07 | Operator spaces and completely bounded maps |