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\(C^*\)-algebras, \(H^*\)-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff and abelian groups

  • Vishvesh Kumar
  • M. W. WongEmail author
Article

Abstract

We define pseudo-differential operators on a locally compact, Hausdorff and abelian group G as natural extensions of pseudo-differential operators on \({\mathbb {R}}^n\). In particular, for pseudo-differential operators with symbols in \(L^2(G\times \widehat{G})\), where \(\widehat{G}\) is the dual group of G, we give explicit formulas for the products and adjoints, characterize them as Hilbert–Schmidt operators on \(L^2(G)\) and prove that they form a \(C^*\)-algebra, which is also a \(H^*\)-algebra. We give a characterization of trace class pseudo-differential operators in terms of symbols lying in a subspace of \(L^1(G\times \widehat{G})\cap L^2(G\times \widehat{G})\).

Keywords

Locally compact Hausdorff and abelian group Haar measure Fourier transform Plancherel formula Fourier inversion formula Pseudo-differential operator Symbol Hilbert–Schmidt operator Trace class operator Trace Product Adjoint \(C^*\)-algebra \(H^*\)-algebra 

Mathematics Subject Classification

Primary 47G30 Secondary 43A32 

Notes

Acknowledgements

The authors are grateful to the referee for pointing out several inaccuracies in the first version of the paper. The first author wishes to thank Council of Scientific and Industrial Research, India, for its research grant. The research of the second author has been supported by Natural Sciences and Engineering Research Council of Canada under Discovery Grant 0008562.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiDelhiIndia
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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