\(K\)-groups of \(C^*\)-algebras deformed by actions of \(\mathbb{R}^ d\)
type: Item
instance of: scholarly article
title: \(K\)-groups of \(C^*\)-algebras deformed by actions of \(\mathbb{R}^ d\)
zbMATH Open document ID: 0803.46077
author: Marc A. Rieffel
published in: Journal of Functional Analysis
publication date: 1994-04-12T00:00:00Z
review text: Earlier [Mem. Am. Math. Soc., to appear] the author has shown, how to deform the original product on the dense \(*\)-subalgebra \(A^ \infty\) of smooth vectors in \(C^*\)-algebra \(A\) with the action \(\alpha\) of \(\mathbb{R}^ d\) upon \(A\). The new associative product is defined by \[ a\times_ J b= \iint \alpha_{Ju}(a) \alpha_ v(b)\exp(2\pi i(u\cdot v)), \] where \(J\) is an arbitrary skew-symmetric operator on \(\mathbb{R}^ d\), \(a,b\in A^ \infty\) and the integral is defined as an oscillatory integral. (In other papers, the author shows how to construct some quantum groups by this process.) The main result of the present paper is that the (usual) topological \(K\)- groups \(K_ j A_ J\) and \(K_ j A\) are isomorphic for \(j= 0,1\). The main step in the proof consists in showing that \(A_ J\) is a generalized fixed-point algebra for a suitable action -- this implies that \(A_ J\) is strongly Morita equivalent to a certain crossed product algebra for an action of \(\mathbb{R}^ d\). Then Connes' Thom isomorphism theorem is invoked to conclude the proof. The main theorem applies to show that the quantum versions of various spaces have the same \(K\)-groups as the original spaces.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L85
zbMATH DE Number: 515248
zbMATH Keywords: deformation
zbMATH Keywords: quantization
zbMATH Keywords: crossed product
zbMATH Keywords: \(\mathbb{R}^ d\)-action
zbMATH Keywords: smooth vectors
zbMATH Keywords: skew-symmetric operator
zbMATH Keywords: oscillatory integral
zbMATH Keywords: quantum groups
zbMATH Keywords: topological \(K\)-groups
zbMATH Keywords: Connes' Thom isomorphism theorem
MaRDI profile type: Publication
full work available at URL: jfan.1993.1110
OpenAlex ID: W1995301934
DOI: 10.1006/JFAN.1993.1110
description: scientific article; zbMATH DE number 515248