\(C^*\)-algebras of singular integral operators with shifts having the same nonempty set of fixed points
type: Item
instance of: scholarly article
title: \(C^*\)-algebras of singular integral operators with shifts having the same nonempty set of fixed points
zbMATH Open document ID: 1158.47066
author: Claudio A. Fernandes
author: Yuri I. Karlovich
author: M. Amélia Bastos
published in: Complex Analysis and Operator Theory
publication date: 2008-09-04T00:00:00Z
review text: Let \(\mathbb{T}\) be the unit circle and \(B(L^2(\mathbb{T}))\) the \(C^*\)-algebra of all bounded linear operators acting on \(L^2(\mathbb{T})\). Consider the \(C^*\)-algebra \(U\subset B(L^2(\mathbb{T}))\) generated by all multiplication operators by piecewise slowly oscillating functions, and by Cauchy singular integral operators. Let \(G\) be an amenable discrete group of orientation-preserving diffeomorphisms of \(\mathbb{T}\) onto intself. The authors consider the \(C^*\)-algebra \(V= \text{alg}(U,G)\). The case when all the shifts \(g\in G\), \(g\neq \text{id}\), have the same finite set of fixed points was studied by the authors in [J.~Funct.\ Anal.\ 242, No.\,1, 86--126 (2007; Zbl 1108.47012)]. Here, the authors allow arbitrary common set of fixed points, with possibly nonempty interior. A symbolic calculus is given with an explicit characterization of Fredholm operators.
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47L15
Mathematics Subject Classification ID: 47A67
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5320875
zbMATH Keywords: Cauchy singular integral operators
zbMATH Keywords: symbolic calculus
zbMATH Keywords: shift operators
zbMATH Keywords: amenable group
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: local-trajectory method
zbMATH Keywords: representation
zbMATH Keywords: spectral measure
zbMATH Keywords: symbol
zbMATH Keywords: Fredholmness
MaRDI profile type: Publication
full work available at URL: s11785-008-0051-4
OpenAlex ID: W2043648201
DOI: 10.1007/S11785-008-0051-4
description: scientific article; zbMATH DE number 5320875