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Annales Henri Poincaré

, Volume 16, Issue 5, pp 1283–1306 | Cite as

Inverse-Closed Algebras of Integral Operators on Locally Compact Groups

  • Ingrid BeltiţăEmail author
  • Daniel Beltiţă
Article

Abstract

We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make use of covariance algebras associated to C*-dynamical systems defined by the C*-algebras of right uniformly continuous functions with respect to the left regular representation.

Keywords

Integral Operator Compact Group Banach Algebra Left Ideal Integral Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Baskakov, A.G.: Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis. (Russian) Sibirsk. Mat. Zh. 38(1), 14–28 (1997) (i; translation in Siberian Math. J. 38(1), 10–22 (1997))Google Scholar
  2. 2.
    Beltiţă I., Beltiţă D.: Algebras of symbols associated with the Weyl calculus for Lie group representations. Monatsh. Math. 167(1), 13–33 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beltiţă I., Beltiţă D.: On the differentiable vectors for contragredient representations. C. R. Math. Acad. Sci. Paris 351(13–14), 513–516 (2013)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Biller H.: Continuous inverse algebras with involution. Forum Math. 22(6), 1033–1059 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Blasco O., Calabuig J.M.: Vector-valued functions integrable with respect to bilinear maps. Taiwan. J. Math. 12(9), 2387–2403 (2008)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Buchholz D., Grundling H.: The resolvent algebra: a new approach to canonical quantum systems. J. Funct. Anal. 254(11), 2725–2779 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Buchholz D., Grundling H.: Lie algebras of derivations and resolvent algebras. Comm. Math. Phys. 320(2), 455–467 (2013)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Comfort W.W., Ross K.A.: Pseudocompactness and uniform continuity in topological groups. Pac. J. Math. 16, 483–496 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Damak M., Georgescu V.: Self-adjoint operators affiliated to C*-algebras. Rev. Math. Phys. 16(2), 257–280 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ellis H.W.: A note on Banach function spaces. Proc. Am. Math. Soc. 9, 75–81 (1958)CrossRefzbMATHGoogle Scholar
  11. 11.
    Ellis H.W., Halperin I.: Function spaces determined by a levelling length function. Can. J. Math. 5, 576–592 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Farrell B., Strohmer T.: Inverse-closedness of a Banach algebra of integral operators on the Heisenberg group. J. Oper. Theory 64(1), 189–205 (2010)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Feichtinger, H.G.: Banach convolution algebras of Wiener type. In: Functions, Series, Operators (Budapest, 1980), vols. I–II, Colloq. Math. Soc. János Bolyai, 35. North-Holland, Amsterdam, pp. 509–524 (1983)Google Scholar
  14. 14.
    Fell, J.M.G., Doran, R.S.: Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, vol. 1. Pure and Applied Mathematics, 125. Academic Press, Inc., Boston (1988)Google Scholar
  15. 15.
    Fendler G., Gröchenig K., Leinert M.: Convolution-dominated operators on discrete groups. Integral Equ. Oper. Theory 61(4), 493–509 (2008)CrossRefzbMATHGoogle Scholar
  16. 16.
    Georgescu V.: On the structure of the essential spectrum of elliptic operators on metric spaces. J. Funct. Anal. 260(6), 1734–1765 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Georgescu V., Iftimovici A.: Crossed products of C*-algebras and spectral analysis of quantum Hamiltonians. Comm. Math. Phys. 228(3), 519–560 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gérard C., Martinez A., Sjöstrand J.: A mathematical approach to the effective Hamiltonian in perturbed periodic problems. Comm. Math. Phys. 142(2), 217–244 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar
  19. 19.
    Girardi M., Weis L.: Integral operators with operator-valued kernels. J. Math. Anal. Appl. 290(1), 190–212 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)Google Scholar
  21. 21.
    Halperin I.: Function spaces. Can. J. Math. 5, 273–288 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hulanicki A.: On the spectrum of convolution operators on groups with polynomial growth. Invent. Math. 17, 135–142 (1972)CrossRefADSzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kister J.M.: Uniform continuity and compactness in topological groups. Proc. Am. Math. Soc. 13, 37–40 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Kurbatov, V.G.: Functional-Differential Operators and Equations. Mathematics and its Applications, vol. 473. Kluwer, Dordrecht (1999)Google Scholar
  25. 25.
    Kurbatov V.G.: Some algebras of operators majorized by a convolution. Funct. Differ. Equ. 8(3–4), 323–333 (2001)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Leptin H.: Darstellungen verallgemeinerter L 1-Algebren. Invent. Math. 5, 192–215 (1968)CrossRefADSzbMATHMathSciNetGoogle Scholar
  27. 27.
    Leptin H.: On symmetry of some Banach algebras. Pac. J. Math. 53, 203–206 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Leptin H., Poguntke D.: Symmetry and nonsymmetry for locally compact groups. J. Funct. Anal. 33(2), 119–134 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Neeb K.-H.: On differentiable vectors for representations of infinite dimensional Lie groups. J. Funct. Anal. 259(11), 2814–2855 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Palmer, Th.W.: Banach Algebras and the General Theory of *-Algebras, vol. 2. Encyclopedia of Mathematics and its Applications, 79. Cambridge University Press, Cambridge (2001)Google Scholar
  31. 31.
    Paterson, A.L.T.: Amenability. Mathematical Surveys and Monographs, 29. American Mathematical Society, Providence (1988)Google Scholar
  32. 32.
    Pedersen, G.K.: C*-algebras and Their Automorphism Groups. London Mathematical Society Monographs, 14. Academic Press, Inc., London (1979)Google Scholar
  33. 33.
    Poguntke D.: Rigidly symmetric L 1-group algebras. Sem. Sophus Lie 2(2), 189–197 (1992)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Rabinovich V.S., Roch S.: The essential spectrum of Schrdinger operators on lattices. J. Phys. A 39(26), 8377–8394 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  35. 35.
    Takai H.: On a duality for crossed products of C*-algebras. J. Funct. Anal. 19, 25–39 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Vallin J.-M.: C*-algèbres de Hopf et C*-algèbres de Kac. Proc. Lond. Math. Soc. (3) 50(1), 131–174 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Williams, D.P.: Crossed Products of C*-Algebras. Mathematical Surveys and Monographs, 134. American Mathematical Society, Providence, RI (2007)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

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