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Invertibility Criteria in \({\varvec{C^*}}\)-algebras of Functional Operators with Shifts and \({\varvec{PQC}}\) Coefficients

  • M. A. BastosEmail author
  • C. A. Fernandes
  • Yu. I. Karlovich
Article
  • 11 Downloads

Abstract

Let G be an amenable discrete group of orientation-preserving piecewise smooth homeomorphisms \(g:\mathbb {T}\rightarrow \mathbb {T}\), with finite sets of discontinuities for their derivatives \(g'\), which acts topologically freely on \(\mathbb {T}{\setminus }\Lambda ^\circ \), where \(\Lambda ^\circ \) is the interior of a nonempty closed set \(\Lambda \subset \mathbb {T}\) composed by all common fixed points for all shifts \(g\in G\). Invertibility criteria are established for the operators in the \(C^*\)-algebra
$$\begin{aligned} {\mathcal {A}}:=\mathrm{alg}\,(PQC,U_G)\subset {\mathcal {B}}(L^2(\mathbb {T})) \end{aligned}$$
generated by all functional operators of the form \(\sum _{g\in F}a_gU_g\), where \(a_gI\) are multiplication operators by piecewise quasicontinuous functions \(a_g\in PQC\) on \(\mathbb {T}\), \(U_g:\varphi \mapsto |g'|^{1/2} (\varphi \circ g)\) are unitary weighted shift operators on \(L^2(\mathbb {T})\), and F is any finite subset of the group G.

Keywords

Amenable group Shift Piecewise quasicontinuous function Functional operator \(C^*\)-algebra Invertibility 

Mathematics Subject Classification

Primary 39B32 Secondary 47B33 47B38 47C15 47L40 

Notes

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Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  2. 2.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaMonte de CaparicaPortugal
  3. 3.Centro de Investigación en Ciencias, Instituto de Investigación en Ciencias Básicas y AplicadasUniversidad Autónoma del Estado de MorelosCuernavacaMexico

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