Skip to main content
SpringerLink
Log in

Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices

  • Published:
Constructive Approximation Aims and scope

Abstract

We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra \(\ensuremath {\mathcal {A}}\), both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on \(\ensuremath {\mathcal {A}}\) and yields a family of smooth inverse-closed subalgebras of \(\ensuremath {\mathcal {A}}\) that resemble the usual Hölder–Zygmund spaces. The second construction starts with a graded sequence of subspaces of \(\ensuremath{\mathcal{A}}\) and yields a class of inverse-closed subalgebras that resemble the classical approximation spaces. We prove a theorem of Jackson–Bernstein type to show that in certain cases both constructions are equivalent.

These results about abstract Banach algebras are applied to algebras of infinite matrices with off-diagonal decay. In particular, we obtain new and unexpected conditions of off-diagonal decay that are preserved under matrix inversion.

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. Almira, J.M., Luther, U.: Approximation algebras and applications. In: Trends in Approximation Theory, Nashville, TN, 2000. Innov. Appl. Math., pp. 1–10. Vanderbilt Univ. Press, Nashville (2001)

    Google Scholar 

  2. Almira, J.M., Luther, U.: Inverse closedness of approximation algebras. J. Math. Anal. Appl. 314(1), 30–44 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Baskakov, A.G.: Wiener’s theorem and asymptotic estimates for elements of inverse matrices. Funktsional. Anal. Prilozhen. 24(3), 64–65 (1990)

    MathSciNet  Google Scholar 

  4. Baskakov, A.G.: Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis. Sib. Mat. Zh. 38(1), 14–28 (1997)

    MathSciNet  Google Scholar 

  5. Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44(1), 45–57 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976). Grundlehren der Mathematischen Wissenschaften No. 223

    MATH  Google Scholar 

  7. Blackadar, B.: K-Theory for Operator Algebras, 2nd edn. Mathematical Sciences Research Institute Publications, vol. 5. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  8. Blackadar, B., Cuntz, J.: Differential Banach algebra norms and smooth subalgebras of C *-algebras. J. Oper. Theory 26(2), 255–282 (1991)

    MATH  MathSciNet  Google Scholar 

  9. Brandenburg, L.: On identifying the maximal ideals in Banach algebras. J. Math. Anal. Appl. 50, 489–510 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bratteli, O.: Inductive limits of finite dimensional C *-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)

    MATH  MathSciNet  Google Scholar 

  11. Bratteli, O.: Derivations, Dissipations and, Group Actions on C *-Algebras. Lecture Notes in Mathematics, vol. 1229. Springer, Berlin (1986)

    Google Scholar 

  12. Bratteli, O., Robinson, D.W.: Unbounded derivations of C *-algebras. Commun. Math. Phys. 42, 253–268 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bratteli, O., Robinson, D.W.: Unbounded derivations of C *-algebras. II. Commun. Math. Phys. 46(1), 11–30 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, 2nd edn. Texts and Monographs in Physics, vol. 1. Springer, New York (1987). C *- and W *-algebras, symmetry groups, decomposition of states

    MATH  Google Scholar 

  15. Butzer, P.L., Berens, H.: Semi-Groups of Operators and Approximation. Die Grundlehren der mathematischen Wissenschaften, vol. 145. Springer, New York (1967)

    MATH  Google Scholar 

  16. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation. Academic Press, New York (1971). Volume 1: One-Dimensional Theory, Pure and Applied Mathematics, Vol. 40

    MATH  Google Scholar 

  17. Butzer, P.L., Scherer, K.: Approximationsprozesse und Interpolationsmethoden. B.I. Hochschulskripten 826/826a. Bibligraphisches Institut, Mannheim (1968)

  18. Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  19. DeLeeuw, K.: Fourier series of operators and an extension of the F. and M. Riesz theorem. Bull. Am. Math. Soc. 79, 342–344 (1973)

    Article  MathSciNet  Google Scholar 

  20. DeLeeuw, K.: An harmonic analysis for operators. I: Formal properties. Illinois J. Math. 19, 593–606 (1975)

    MATH  MathSciNet  Google Scholar 

  21. DeLeeuw, K.: An harmonic analysis for operators. II: Operators on Hilbert space and analytic operators. Illinois J. Math. 21, 164–175 (1977)

    MATH  MathSciNet  Google Scholar 

  22. Demko, S., Moss, W.F., Smith, P.W.: Decay rates for inverses of band matrices. Math. Comput. 43(168), 491–499 (1984)

    MATH  MathSciNet  Google Scholar 

  23. DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 303. Springer, Berlin (1993)

    MATH  Google Scholar 

  24. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer-Verlag, New York (2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

    MATH  Google Scholar 

  25. Fendler, G., Gröchenig, K., Leinert, M.: Symmetry of weighted L 1-algebras and the GRS-condition. Bull. Lond. Math. Soc. 38(4), 625–635 (2006)

    Article  MATH  Google Scholar 

  26. Glimm, J.G.: On a certain class of operator algebras. Trans. Am. Math. Soc. 95, 318–340 (1960)

    MATH  MathSciNet  Google Scholar 

  27. Gohberg, I., Kaashoek, M.A., Woerdeman, H.J.: The band method for positive and strictly contractive extension problems: an alternative version and new applications. Integral Equ. Oper. Theory 12(3), 343–382 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gröchenig, K.: Wiener’s Lemma: Theme and Variations. An Introduction to Spectral Invariance. Four Short Courses on Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2010)

    Google Scholar 

  29. Gröchenig, K., Leinert, M.: Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices. Trans. Am. Math. Soc. 358(6), 2695–2711 (2006) (electronic)

    Article  MATH  Google Scholar 

  30. Gröchenig, K., Rzeszotnik, Z.: Almost diagonalization of pseudodifferential operators. Ann. Inst. Fourier 58(6), 2279–2314 (2008)

    MATH  Google Scholar 

  31. Grushka, Y., Torba, S.: Direct theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors. Methods Funct. Anal. Topol. 13(3), 267–278 (2007)

    MATH  MathSciNet  Google Scholar 

  32. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence (1957), rev. edn.

    Google Scholar 

  33. Hulanicki, A.: On the spectrum of convolution operators on groups with polynomial growth. Invent. Math. 17, 135–142 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  34. Jaffard, S.: Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(5), 461–476 (1990)

    MATH  MathSciNet  Google Scholar 

  35. Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  36. Kissin, E., Shul’man, V.S.: Dense Q-subalgebras of Banach and C *-algebras and unbounded derivations of Banach and C *-algebras. Proc. Edinb. Math. Soc. (2) 36(2), 261–276 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kissin, E., Shul’man, V.S.: Differential properties of some dense subalgebras of C *-algebras. Proc. Edinb. Math. Soc. (2) 37(3), 399–422 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  38. Klotz, A.: Noncommutative approximation: smoothness and approximation and invertibility in Banach algebras. PhD thesis, University of Vienna (2010)

  39. Krantz, S.G.: Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math. 1(3), 193–260 (1983)

    MATH  MathSciNet  Google Scholar 

  40. Kurbatov, V.G.: Algebras of difference and integral operators. Funktsional. Anal. Prilozhen. 24(2), 87–88 (1990)

    MathSciNet  Google Scholar 

  41. Nikolski, N.: In search of the invisible spectrum. Ann. Inst. Fourier (Grenoble) 49(6), 1925–1998 (1999)

    MathSciNet  Google Scholar 

  42. Pietsch, A.: Approximation spaces. J. Approx. Theory 32(2), 115–134 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  43. Rabinovich, V.S., Roch, S., Silbermann, B.: Fredholm theory and finite section method for band-dominated operators. Integral Equ. Oper. Theory 30(4), 452–495 (1998). Dedicated to the memory of Mark Grigorievich Krein (1907–1989)

    Article  MATH  MathSciNet  Google Scholar 

  44. Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory. Operator Theory: Advances and Applications, vol. 150. Birkhäuser, Basel (2004)

    MATH  Google Scholar 

  45. Sjöstrand, J.: Wiener type algebras of pseudodifferential operators. In: Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, pages Exp. No. IV, p. 21. École Polytech., Palaiseau (1995)

    Google Scholar 

  46. Sun, Q.: Wiener’s lemma for infinite matrices. Trans. Am. Math. Soc. 359(7), 3099–3123 (2007)

    Article  MATH  Google Scholar 

  47. Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, vol. 34. Pergamon/Macmillan, New York (1963)

    MATH  Google Scholar 

  48. Torba, S.: Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors. arXiv:0809.4030v1 [math.FA] (2008)

  49. Trigub, R.M., Bellinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer Academic, Dordrecht (2004)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Vienna, Austria

    Karlheinz Gröchenig & Andreas Klotz

Authors
  1. Karlheinz Gröchenig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Klotz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karlheinz Gröchenig.

Additional information

Communicated by Albert Cohen.

K.G. was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154. A.K.  was supported by the grant MA44 MOHAWI of the Vienna Science and Technology Fund (WWTF) and partially by the National Research Network S106 SISE of the Austrian Science Foundation (FWF).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gröchenig, K., Klotz, A. Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices. Constr Approx 32, 429–466 (2010). https://doi.org/10.1007/s00365-010-9101-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-010-9101-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation