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Local and Analytic Cyclic Homology
Ralf Meyer, University of Göttingen, Germany
A publication of the European Mathematical Society.
cover
EMS Tracts in Mathematics
2007; 368 pp; hardcover
Volume: 3
ISBN-10: 3-03719-039-6
ISBN-13: 978-3-03719-039-5
List Price: US$78
Member Price: US$62.40
Order Code: EMSTM/3
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Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book, the author develops and compares these theories, emphasizing their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to \(K\)-theory, and the Chern-Connes character for \(K\)-theory and \(K\)-homology.

The cyclic homology theories studied in this text require a good deal of functional analysis in bornological vector spaces, which is supplied in the first chapters. The focal points here are the relationship with inductive systems and the functional calculus in non-commutative bornological algebras.

Some chapters are more elementary and independent of the rest of the book and will be of interest to researchers and students working on functional analysis and its applications.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in local and analytic cyclic homology.

Table of Contents

  • Bornological vector spaces and inductive systems
  • Relations between entire, analytic, and local cyclic homology
  • The spectral radius of bounded subsets and its applications
  • Periodic cyclic homology via pro-nilpotent extensions
  • Analytic cyclic homology and analytically nilpotent extensions
  • Local homotopy invariance and isoradial subalgebras
  • The Chern-Connes character
  • Appendix. Background material
  • Bibliography
  • Notation and symbols
  • Index
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