Cyclic homology in non-commutative geometry
type: Item
instance of: scholarly article
title: Cyclic homology in non-commutative geometry
zbMATH Open document ID: 1045.46043
author: Joachim Cuntz
author: Georges Skandalis
author: Boris Tsygan
published in: Encyclopaedia of Mathematical Sciences
publication date: 2003-12-03T00:00:00Z
review text: This book consists of the following three parts: ``Cyclic theory, bivariant \(K\)-Theory and the bivariant Chern-Connes character'' by J.~Cuntz; ``Cyclic homology'' by B.~Tsygan; and ``Noncommutative geometry, the transverse signature operator, and Hopf algebras (after A.~Connes and H.~Moscovici)'' by \textit{G.~Skandalis} (translation of [Astérisque 282, 345--364 (2002; Zbl 1035.58007)] by R.~Ponge and N.~Wright). They give complete accounts of cyclic theory from different and complementary points of view. The first and the second parts both deal with cyclic theory. But the first part aimed to present the bivariant Chern--Connes character on the category of \(m\)-algebras (algebras that can be represented as projective limits of Banach algebras), and infinite-dimensional cyclic theory [\textit{J.~Cuntz}, Doc. Math. J. DMV 2, 139--182 (1997; Zbl 0920.19004), hereafter referred to as [1], \textit{R.~Meyer}, J. Eur. Math. Soc. (JEMS) 3, 269--286 (2001; Zbl 0990.19002)]. While the second part aimed to present Tamarkin--Tsygan's noncommutative differential calculus and applications of cyclic theory to the deformation quantization, especially to the index theorem for deformation [\textit{D.~Tamarkin} and \textit{B.~Tsygan}, Lett. Math. Phys. 56, 85--97 (2001; Zbl 1008.19002); hereafter referred to as [2], \textit{R.~Nest} and \textit{B.~Tsygan}, Commun. Math. Phys. 172, 223--262 (1995; Zbl 0887.58050), hereafter referred to as [3], \textit{B.~V.~Fedosov}, Math. Top. 5, 277-297 (1994; Zbl 0809.58012), hereafter referred to as [4]]. The third part is aimed to expose the local index theorem of \textit{A.~Connes} and \textit{H.~Moscovici} [Geom. Funct. Anal. 5, 174--243 (1995; Zbl 0960.46048), hereafter referred to as [5], Commun. Math. Phys, 48, 97--108 (1999; Zbl 0941.16024), hereafter referred to as [6]]. Besides the introduction, the first part consists of four chapters entitled ``2. Cyclic Theory'', ``3. Cyclic Theory for Locally Convex Algebras'', ``4. Bivariant \(K\)-…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 46-00
zbMATH DE Number: 2012369
zbMATH Keywords: cyclic bicomplex
zbMATH Keywords: Connes complex
zbMATH Keywords: \(X\)-complex
zbMATH Keywords: bivariant \(K\)-theory
zbMATH Keywords: noncommutative calculus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2012369