Caractère multiplicatif d'un module de Fredholm. (Multiplicative character of a Fredholm module)
type: Item
instance of: scholarly article
title: Caractère multiplicatif d'un module de Fredholm. (Multiplicative character of a Fredholm module)
zbMATH Open document ID: 0671.46034
DOI: 10.1007/BF00533391
author: Max Karoubi
author: Alain Connes
published in: \(K\)-Theory
publication date: 1988-01-01T00:00:00Z
review text: Let A be an algebra. An odd Fredholm module is an action \(\rho\) of A on a \({\mathbb{Z}}/2\)-graded Hilbert space \(H=H^+\oplus H^-\) such that the off- diagonal elements of \(\rho\) (a), \(a\in A\) are compact. An even Fredholm module consists of two actions \(\rho_ 1\), \(\rho_ 2\) of A on a Hilbert space such that \(\rho_ 1(a)-\rho_ 2(a)\), \(a\in A\) is compact. If the off-diagonal elements or these differences are not only compact but even in the Schatten ideal \(\ell^ p\) one says that the module is p-summable. The module is said to be of dimension p if it is \((p+1)\)-summable and of the same parity as p. It is well known (by previous results of the first author) that any Fredholm module of dimension p induces additive homomorphisms \(K_ i(A)\to {\mathbb{Z}}\subset {\mathbb{C}}\) where \(i=0,1\) according to the parity of p. In the present article the authors show that such a module also induces a multiplicative homomorphism \(K_{p+1}(A)\to {\mathbb{C}}^*\) where \(K_ n\) is the algebraic K-theory in the sense of Quillen. It suffices, obviously, to construct this homomorphism for the universal algebra \({\mathcal M}^ p\) connected with a universal Fredholm module of dimension p. This algebra is a Banach algebra and one obtains the desired homomorphism from an exact sequence connecting the algebraic and topological K-theory using certain canonical cyclic cocycles on \({\mathcal M}^ p.\) Finally the authors describe the central extensions of SL(\({\mathcal C}^{\infty}(S^ 1))\) and of \(Diff^+(S^ 1)\) associated to certain Fredholm modules and their relation to the Virasoro algebra.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 4098061
zbMATH Keywords: even Fredholm module
zbMATH Keywords: Schatten ideal
zbMATH Keywords: algebraic K-theory in the sense of Quillen
zbMATH Keywords: universal algebra
zbMATH Keywords: universal Fredholm module of dimension p
zbMATH Keywords: homomorphism
zbMATH Keywords: exact sequence
zbMATH Keywords: Virasoro algebra
Wikidata QID: Q104147077
reviewed by: Joachim Cuntz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00533391
OpenAlex ID: W2053074542
description: scientific article; zbMATH DE number 4098061