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If mathematics is a language, then taking a topology course at the
undergraduate level is cramming vocabulary and memorizing irregular
verbs: a necessary, but not always exciting exercise one has to go
through before one can read great works of literature in the
original language.The present book grew out of notes for an
introductory topology course at the University of Alberta. It
provides a concise introduction to set theoretic topology (and to a
tiny little bit of algebraic topology). It is accessible to
undergraduates from the second year on, but even beginning graduate
students canbenefit from some parts.Great care has been devoted to
the selection of examples that are not self-serving, but already
accessible for students who have a background in calculus and
elementary algebra, but not necessarily in real or complex
analysis.In some points, the book treats its material differently
than other texts on the subject:
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.
This volume provides readers with a detailed introduction to the
amenability of Banach algebras and locally compact groups. By
encompassing important foundational material, contemporary
research, and recent advancements, this monograph offers a
state-of-the-art reference. It will appeal to anyone interested in
questions of amenability, including those familiar with the
author's previous volume Lectures on Amenability. Cornerstone
topics are covered first: namely, the theory of amenability, its
historical context, and key properties of amenable groups. This
introduction leads to the amenability of Banach algebras, which is
the main focus of the book. Dual Banach algebras are given an
in-depth exploration, as are Banach spaces, Banach homological
algebra, and more. By covering amenability's many applications, the
author offers a simultaneously expansive and detailed treatment.
Additionally, there are numerous exercises and notes at the end of
every chapter that further elaborate on the chapter's contents.
Because it covers both the basics and cutting edge research,
Amenable Banach Algebras will be indispensable to both graduate
students and researchers working in functional analysis, harmonic
analysis, topological groups, and Banach algebras. Instructors
seeking to design an advanced course around this subject will
appreciate the student-friendly elements; a prerequisite of
functional analysis, abstract harmonic analysis, and Banach algebra
theory is assumed.
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