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Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 3018–3044 | Cite as

Norm-Controlled Inversion in Weighted Convolution Algebras

  • Ebrahim SameiEmail author
  • Varvara Shepelska
Article
  • 26 Downloads

Abstract

Let G be a discrete group, let \(p\ge 1\), and let \(\omega \) be a weight on G. Using the approach from Gröchenig and Klotz (J Lond Math Soc (2) 88:49–64, 2013), we provide sufficient conditions on a weight \(\omega \) for \(\ell ^p(G,\omega )\) to be a Banach algebra admitting a norm-controlled inversion in \(C^*_r(G)\). We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in \(B(L^2(G))\) for some related convolution algebras.

Keywords

Norm-controlled inversion Locally compact groups Convolution algebras Weights Groups of polynomial growth 

Mathematics Subject Classification

43A10 43A15 47A60 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for the numerous suggestions that helped to improve the presentation of the paper. In particular, the current proof of Lemma 5.8 is due to a referee and is much less technical than the original one. The first name author would also like to thank the great hospitality he received from IMPAN during his stay.

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada
  2. 2.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland

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