Twisted dual-group algebras: Equivariant deformations of \(C_ 0(G)\)
type: Item
instance of: scholarly article
title: Twisted dual-group algebras: Equivariant deformations of \(C_ 0(G)\)
zbMATH Open document ID: 0839.22003
author: Magnus B. Landstad
author: Iain Raeburn
published in: Journal of Functional Analysis
publication date: 1996-06-19T00:00:00Z
review text: ``We consider a family of twisted Fourier algebras \(A(G, \omega)\) of a locally compact group \(G\), which in the case of an abelian group \(G\) are the Fourier transforms of the usual twisted group algebras of \(\widehat G\). The corresponding \(C^*\)-algebras \(C^* (\widehat G, \omega)\) are deformations of \(C_0 (G)\), which are equivariant in the sense that \(G\) still acts by left translation. The main examples come from cocycles \(\sigma\) on the dual of an abelian subgroup \(H\) of \(G\); we prove that for such cocycles the twisted dual-group algebras \(C^* (\widehat G, \omega)\) are induced from the twisted group algebras \(C^* (\widehat H, \sigma)\), and we give detailed formulas for the multiplication on \(A(G, \omega)\) which extend to larger dense subalgebras of \(C_0 (G)\) and \(C_b (G)\). We anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces \(C_0 (G/ \Gamma)\).'' The formulation goes like this. Let \({\mathcal L} (G)\) be the group von Neumann algebra (generated by the left regular representation), and let \(\delta_G : {\mathcal L} (G) \to {\mathcal L} (G) \otimes {\mathcal L} (G)\) be the unique normal homomorphism such that \(\delta_G (L_x) = L_x \otimes L_x\) on the left translation operators. Then a dual cocycle is a unitary \(\omega \in {\mathcal L} (G) \otimes {\mathcal L} (G)\) satisfying \[ \omega_{12} \bigl( \delta_G \otimes i (\omega) \bigr) = \omega_{23} \bigl( i \otimes \delta_G (\omega) \bigr) \quad \text{in } {\mathcal L} (G) \otimes {\mathcal L} (G) \otimes {\mathcal L} (G) \] (which, for abelian \(G\), is the image of the cocycle identity under the Fourier transform of \({\mathcal L} (G)\) onto \(L^\infty (\widehat G))\). For \(f,g \in A (G)\), the twisted multiplication \(f*_\omega g\) in the Fourier algebra \(A(G)\) is defined by \[ \langle T, f*_\omega g \rangle = \bigl \langle \delta_G (T) \omega^*, f \otimes g \bigr \rangle \quad \text{for } T \in {\mathcal L} (G). \]
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L99
Mathematics Subject Classification ID: 43A25
zbMATH DE Number: 790374
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: locally compact group
zbMATH Keywords: abelian group
zbMATH Keywords: Fourier transforms
zbMATH Keywords: group algebras
zbMATH Keywords: cocycles
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: left regular representation
zbMATH Keywords: left translation operators
reviewed by: Paul Milnes
MaRDI profile type: Publication
full work available at URL: jfan.1995.1100
OpenAlex ID: W1984590158
DOI: 10.1006/JFAN.1995.1100
description: scientific article; zbMATH DE number 790374