# Local and Analytic Cyclic Homology

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*Ralf Meyer*

A publication of the European Mathematical Society

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.