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Local and analytic cyclic homology. (English) Zbl 1134.46001
EMS Tracts in Mathematics 3. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-039-5/hbk). viii, 360 p. (2007).
This monograph deals with two closely related variants of periodic cyclic homology, called analytic cyclic homology \(HA_*\) and local cyclic homology \(HL_*\), the first of which was introduced by the author in his doctoral thesis based on the entire cyclic cohomology of A. Connes. There he observed the relevance of bornological vector spaces and studied the formal properties of \(HA_*\). The present monograph is based on the author’s thesis but rewritten almost entirely. The main change is the inclusion of bivariant local cyclic homology, which is quite close to the analytic theory, but has much better formal properties, with the main difference being that complete bornological vector spaces are replaced by inductive systems of Banach spaces. The detailed exposition of the analytic and local cyclic homology is concluded by the description of the bivariant Chern-Connes character as a map from Kasparov’s \(KK\)-theory to \(HL_*\).

MSC:
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
19K35 Kasparov theory (\(KK\)-theory)
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