Skip to main content
SpringerLink
Log in

Derivations on the algebra of Rajchman measures

  • RESEARCH
  • Published:
Complex Analysis and its Synergies Aims and scope Submit manuscript

Abstract

For a locally compact Abelian group G, the algebra of Rajchman measures, denoted by \({M}_{0}(G)\), is the set of all bounded regular Borel measures on G Fourier transform of which vanish at infinity. In this paper, we investigate the spectral structure of the algebra of Rajchman measures, and illustrate aspects of the residual analytic structure of its maximal ideal space. In particular, we show that \({M}_{0}(G)\) has a nonzero continuous point derivation, whenever G is a nondiscrete locally compact Abelian group. We then give the definition of the Rajchman algebra for a general (not necessarily Abelian) locally compact group, and prove that for a noncompact connected SIN group, the Rajchman algebra admits a nonzero continuous point derivation. Moreover, we discuss the analytic behavior of the spectrum of \({M}_{0}(G)\). Namely, we show that for every nondiscrete metrizable locally compact Abelian group G, the maximal ideal space of \({M}_{0}(G)\) contains analytic disks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bade, W.G., Curtis Jr., P.C., Dales, H.G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. Lond. Math. Soc. (3) 55(2), 359–377 (1987)

    Article  MathSciNet  Google Scholar 

  2. Baggett, L., Taylor, K.F.: Riemann-Lebesgue subsets of ${ R}^{n}$ and representations which vanish at infinity. J. Funct. Anal. 28(2), 168–181 (1978)

    Article  MathSciNet  Google Scholar 

  3. Baggett, L., Taylor, K.: A sufficient condition for the complete reducibility of the regular representation. J. Funct. Anal. 34(2), 250–265 (1979)

    Article  MathSciNet  Google Scholar 

  4. Baggett, L., Taylor, K.F.: On asymptotic behavior of induced representations. Can. J. Math. 34(1), 220–232 (1982)

    Article  MathSciNet  Google Scholar 

  5. Bourbaki, N.: Éléments de mathématique. Fasc. XXXV. Livre VI: Intégration. Chapitre IX: Intégration sur les espaces topologiques séparés. Actualités Scientifiques et Industrielles, vol. 1343. Hermann, Paris (1969)

    MATH  Google Scholar 

  6. Brown, G., Moran, W.: Point derivations on $M(G)$. Bull. Lond. Math. Soc. 8(1), 57–64 (1976)

    Article  MathSciNet  Google Scholar 

  7. Brown, G., Moran, W.: Analytic discs in the maximal ideal space of $M(G)$. Pac. J. Math. 75(1), 45–57 (1978)

    Article  MathSciNet  Google Scholar 

  8. Choi, Y., Ghandehari, M.: Weak and cyclic amenability for Fourier algebras of connected Lie groups. J. Funct. Anal. 266(11), 6501–653 (2014)

    Article  MathSciNet  Google Scholar 

  9. Choi, Y., Ghandehari, M.: Weak amenability for fourier algebras of 1-connected nilpotent lie groups. J. Funct. Anal. 268(8), 2440–2463 (2015)

    Article  MathSciNet  Google Scholar 

  10. Connes, A.: On the cohomology of operator algebras. J. Funct. Anal. 28(2), 248–253 (1978)

    Article  MathSciNet  Google Scholar 

  11. Dales, H.G., Pandey, S.S.: Weak amenability of Segal algebras. Proc. Am. Math. Soc. 128(5), 1419–1425 (2000)

    Article  MathSciNet  Google Scholar 

  12. Dales, H.G., Ghahramani, F., Helemskii, A.Y.: The amenability of measure algebras. J. Lond. Math. Soc. (2) 66(1), 213–226 (2002)

    Article  MathSciNet  Google Scholar 

  13. Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92, 181–236 (1964)

    Article  MathSciNet  Google Scholar 

  14. Forrest, B., Wood, P.: Cohomology and the operator space structure of the Fourier algebra and its second dual. Indiana Univ. Math. J. 50(3), 1217–1240 (2001)

    Article  MathSciNet  Google Scholar 

  15. Forrest, B.E., Runde, V.: Amenability and weak amenability of the Fourier algebra. Math. Z. 250(4), 731–744 (2005)

    Article  MathSciNet  Google Scholar 

  16. Forrest, B.E., Samei, E., Spronk, N.: Weak amenability of Fourier algebras on compact groups. Indiana Univ. Math. J. 58(3), 1379–1393 (2009)

    Article  MathSciNet  Google Scholar 

  17. Ghandehari, M.: Amenability properties of Rajchman algebras. Indiana Univ. Math. J. 61(3), 1369–1392 (2012)

    Article  MathSciNet  Google Scholar 

  18. Graham, C.C., McGehee, O.C.: Essays in commutative harmonic analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238. Springer, New York (1979)

    Google Scholar 

  19. Haagerup, U.: All nuclear $C^{\ast } $-algebras are amenable. Invent. Math. 74(2), 305–319 (1983)

    Article  MathSciNet  Google Scholar 

  20. Helemskii, A.Y.: Homology of Banach and Topological Algebras. Kluver, Dordrecht (1989)

    Book  Google Scholar 

  21. Howe, R.E., Moore, C.C.: Asymptotic properties of unitary representations. J. Funct. Anal. 32(1), 72–96 (1979)

    Article  MathSciNet  Google Scholar 

  22. Johnson, B.E.: Approximate diagonals and cohomology of certain annihilator Banach algebras. Am. J. Math. 94, 685–698 (1972)

    Article  MathSciNet  Google Scholar 

  23. Johnson, B.E.: Cohomology in Banach Algebras. Memoirs of the American Mathematical Society, vol. 127. American Mathematical Society, Providence (1972)

    Google Scholar 

  24. Johnson, B.E.: Weak amenability of group algebras. Bull. Lond. Math. Soc. 23(3), 281–284 (1991)

    Article  MathSciNet  Google Scholar 

  25. Johnson, B.E.: Non-amenability of the Fourier algebra of a compact group. J. London Math. Soc. 50, 361–374 (1994)

    Article  MathSciNet  Google Scholar 

  26. Kaniuth, E., Lau, A.T.-M., Ülger, A.: The Rajchman algebra $B_0(G)$ of a locally compact group $G$. Bull. Sci. Math. 140(3), 273–302 (2016)

    Article  MathSciNet  Google Scholar 

  27. Knudby, S.: Fourier algebras of parabolic subgroups. Math. Scand. 120(2), 272–290 (2017)

    Article  MathSciNet  Google Scholar 

  28. Knudby, S.: Groups whose Fourier algebra and Rajchman algebra coincide. J. Math. Anal. Appl. 455(2), 1401–1424 (2017)

    Article  MathSciNet  Google Scholar 

  29. Lau, A.T.-M., Loy, R.J.: Weak amenability of Banach algebras on locally compact groups. J. Funct. Anal. 145(1), 175–204 (1997)

    Article  MathSciNet  Google Scholar 

  30. Lee, H.H., Ludwig, J., Samei, E., Spronk, N.: Weak amenability of Fourier algebras and local synthesis of the anti-diagonal. Adv. Math. 292, 11–41 (2016)

    Article  MathSciNet  Google Scholar 

  31. Menshov, D.: Sur l’unicité du développement trigonométrique. C.R. Acad. Sci. Paris 163, 433–436 (1916)

    MATH  Google Scholar 

  32. Moore, C.C.: Ergodicity of flows on homogeneous spaces. Am. J. Math. 88, 154–178 (1966)

    Article  MathSciNet  Google Scholar 

  33. Rudin, W.: Fourier-Stieltjes transforms of measures on independent sets. Bull. Am. Math. Soc. 66, 199–202 (1960)

    Article  MathSciNet  Google Scholar 

  34. Rudin, W.: Fourier Analysis on Groups, 2nd edn. Wiley, New York (1990)

    Book  Google Scholar 

  35. Runde, V.: Lectures on Amenability. Lecture Notes in Mathematics, vol. 1774. Springer, Berlin (2002)

    Book  Google Scholar 

  36. Runde, V., Spronk, N.: Operator amenability of Fourier-Stieltjes algebras. II. Bull. Lond. Math. Soc. 39(2), 194–202 (2007)

    Article  MathSciNet  Google Scholar 

  37. Shintani, T.: On certain square-integrable irreducible unitary representations of some $P$-adic linear groups. J. Math. Soc. Jpn. 20, 522–565 (1968)

    Article  MathSciNet  Google Scholar 

  38. Taylor, J.L.: The structure of convolution measure algebras. Trans. Am. Math. Soc. 119, 150–166 (1965)

    Article  MathSciNet  Google Scholar 

  39. Varopoulos, NTh: A direct decomposition of the measure algebra of a locally compact Abelian group. Ann. Inst. Fourier (Grenoble) 16((fasc. 1)), 121–143 (1966)

    Article  MathSciNet  Google Scholar 

  40. Varopoulos, NTh: Sets of multiplicity in locally compact abelian groups. Ann. Inst. Fourier (Grenoble) 16((fasc. 2)), 123–158 (1966)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research presented in this article was part of the author’s Ph.D. thesis. The author would like to thank my supervisors Brian Forrest and Nico Spronk for their encouragement and invaluable discussions and suggestions. The author offers her many thanks to H. Garth Dales and Colin Graham for their suggestions. At the time this article was written, she was supported by the University of Delaware Research Foundation Grant number, 17A00999.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, USA

    Mahya Ghandehari

Authors
  1. Mahya Ghandehari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahya Ghandehari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghandehari, M. Derivations on the algebra of Rajchman measures. Complex Anal Synerg 5, 6 (2019). https://doi.org/10.1007/s40627-019-0025-5

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40627-019-0025-5

Keywords

Mathematics Subject Classification

Search

Navigation