Excision in entire cyclic cohomology
type: Item
instance of: scholarly article
title: Excision in entire cyclic cohomology
zbMATH Open document ID: 0990.19002
author: Ralf Meyer
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2002-08-07T00:00:00Z
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 46A08
zbMATH DE Number: 1656347
zbMATH Keywords: entire cyclic cohomology
zbMATH Keywords: excision
zbMATH Keywords: bornological algebras
MaRDI profile type: Publication
full work available at URL: s100970100035
OpenAlex ID: W2053513725
DOI: 10.1007/S100970100035
Recommended article: Excision in cyclic homology theories
Recommended article: Excision in bivariant periodic cyclic cohomology
Recommended article: Q4379453
Recommended article: Local and analytic cyclic homology
Recommended article: Cyclic cohomology, supertraces and index theory
Recommended article: On the entire cyclic cohomology of banach algebras
Recommended article: Algebraic connections, universal bimodules and entire cyclic cohomology
Recommended article: Q4465028
Recommended article: Universal extension and excision for topological algebras
Recommended article: Q4896456
review text: The author generalizes Connes' definition of entire cyclic cohomology \(HE^*\) to bornological algebras. By a bornological algebra is meant a (convex) bornological vector space with a bounded associative multiplication. If \(A\) is an algebra, then Cuntz and Quillen's approach to periodic cyclic homology yields a complex which is denoted \(X(\mathcal TA)\). If \(A\) is a bornological algebra the author gives \(X(\mathcal TA)\) a bornology, and defines the entire cyclic cohomology \(HE^*(A)\) as the complex of bounded linear maps \(X(\mathcal TA)\to\mathbb{C}\). NEWLINENEWLINEAs the title indicates, the paper is devoted to showing the following excision result. Let NEWLINE\[NEWLINE0\longrightarrow K\overset{i}\longrightarrow E\overset{p}\longrightarrow Q\longrightarrow 0NEWLINE\] NEWLINEbe an extension of bornological algebras with a bounded linear section \(Q\to E\). Then the bornological complex \(X(\mathcal TE:\mathcal TQ)\) calculating the relative entire cyclic cohomology is naturally homotopy equivalent to \(X(\mathcal TK)\). It follows that there is a natural six-term exact sequence NEWLINE\[NEWLINE\begin{tikzcd} NEWLINEHE^0(Q)\ar[r,"p^\ast"] & HE^0(E)\ar[r,"i^\ast"] & HE^0(K)\ar[d]\\ NEWLINEHE^1(K)\ar[u] & HE^1(E)\ar[l,"i^\ast" '] & HE^1(Q)\ar[l,"p^\ast" ']\rlap{\,.} NEWLINE\end{tikzcd}NEWLINE\]NEWLINEThe proof also yields similar results for the associated bivariant and periodic theories. NEWLINENEWLINEThe proof presented in this paper differs from the proofs of similar statements as by \textit{J. Cuntz} [in: Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun. 17, 43-53 (1997; Zbl 0890.18006)], \textit{J. Cuntz} and \textit{D. Quillen} [Invent. Math. 127, No. 1, 67-98 (1997; Zbl 0889.46054)], \textit{M. Puschnigg} [Invent. Math. 143, No. 2, 249-323 (2001; Zbl 0973.19002)] in that it does not rely on the methods of \textit{M. Wodzicki} [Ann. Math., II. Ser. 129, No. 3, 591-639 (1989; Zbl 0689.16013)] which are ``difficult …
description: scientific article; zbMATH DE number 1656347