Papers
Topics
Authors
Recent
Search
2000 character limit reached

Symmetric functionals on simply generated symmetric spaces

Published 22 Apr 2024 in math.OA and math.FA | (2404.13870v1)

Abstract: In the present paper we suggest a construction of symmetric functionals on a large class of symmetric spaces over a semifinite von Neumann algebra. This approach establishes a bijection between the symmetric functionals on symmetric spaces and shift-invariant functionals on the space of bounded sequences. It allows to obtain a bijection between the classes of all continuous symmetric functionals on different symmetric spaces. Notably, we show that this mapping is not bijective on the class of all Dixmier traces. As an application of our results we prove an extension of the Connes trace formula for a wide class of operators and symmetric functionals.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (44)
  1. Regular variation, vol. 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1989.
  2. Non-commutative residues for anisotropic pseudo-differential operators in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. J. Funct. Anal. 203, 2 (2003), 305–320.
  3. Boos, J. Classical and modern methods in summability. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000. Assisted by Peter Cass, Oxford Science Publications.
  4. Calkin, J. W. Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. of Math. (2) 42 (1941), 839–873.
  5. Connes, A. The action functional in noncommutative geometry. Comm. Math. Phys. 117, 4 (1988), 673–683.
  6. Connes, A. Noncommutative geometry. Academic Press Inc., San Diego, CA, 1994.
  7. Dixmier, J. Existence de traces non normales. C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107–A1108.
  8. Dixmier, J. Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann). Gauthier-Villars Éditeur, Paris, 1969. Deuxième édition, revue et augmentée, Cahiers Scientifiques, Fasc. XXV.
  9. Symmetric functionals and singular traces. Positivity 2, 1 (1998), 47–75.
  10. Noncommutative Integration and Operator Theory. Birkhäuser Cham, 2024.
  11. Commutator structure of operator ideals. Adv. Math. 185, 1 (2004), 1–79.
  12. Weyl’s law and Connes’ trace theorem for noncommutative two tori. Lett. Math. Phys. 103, 1 (2013), 1–18.
  13. Symmetric linear functionals on function spaces. Function spaces, interpolation theory and related topics, (Lund, 2000), de Gruyter, Berlin, 2002, 311–332.
  14. Dixmier traces and residues on weak operator ideals. submitted manuscript (arXiv:1710.08260).
  15. Singular traces on semifinite von Neumann algebras. J. Funct. Anal. 134, 2 (1995), 451–485.
  16. Fundamentals of the theory of operator algebras. Vol. I, vol. 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Elementary theory, Reprint of the 1983 original.
  17. Fundamentals of the theory of operator algebras. Vol. II, vol. 16 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Advanced theory, Corrected reprint of the 1986 original.
  18. Traces of compact operators and the noncommutative residue. Adv. Math. 235 (2013), 1–55.
  19. Pietsch correspondence for symmetric functionals on Calkin operator spaces associated with semifinite von Neumann algebras. J. Operator Theory 89:2 (2023), 305–342.
  20. Singular traces. Vol. 2. Trace formulas, vol. 46/2 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2023.
  21. Measures from Dixmier traces and zeta functions. J. Funct. Anal. 259, 8 (2010), 1915–1949.
  22. Singular Traces: Theory and Applications, vol. 46 of Studies in Mathematics. De Gruyter, 2012.
  23. Lorentz, G. G. A contribution to the theory of divergent sequences. Acta Math. 80 (1948), 167–190.
  24. Meyer-Nieberg, P. Banach lattices. Universitext. Springer-Verlag, Berlin, 1991.
  25. Nelson, E. Notes on non-commutative integration. J. Functional Analysis 15 (1974), 103–116.
  26. Pietsch, A. Traces and shift invariant functionals. Math. Nachr. 145 (1990), 7–43.
  27. Pietsch, A. Traces on operator ideals and related linear forms on sequence ideals (part I). Indag. Math. (N.S.) 25, 2 (2014), 341–365.
  28. Pietsch, A. Traces on operator ideals and related linear forms on sequence ideals (part II). Integral Equations Operator Theory 79, 2 (2014), 255–299.
  29. Pietsch, A. Traces and Residues of Pseudo-Differential Operators on the Torus. Integral Equations Operator Theory 83, 1 (2015), 1–23.
  30. Pietsch, A. Traces on operator ideals and related linear forms on sequence ideals (part III). J. Math. Anal. Appl. 421, 2 (2015), 971–981.
  31. Pietsch, A. A new approach to operator ideals on Hilbert space and their traces. Integral Equations Operator Theory 89, 4 (2017), 595–606.
  32. Pietsch, A. A new view at Dixmier traces on ℒ1,∞⁢(H)subscriptℒ1𝐻\mathcal{L}_{1,\infty}(H)caligraphic_L start_POSTSUBSCRIPT 1 , ∞ end_POSTSUBSCRIPT ( italic_H ). Integral Equations Operator Theory 91, 3 (2019), Art. 21, 29.
  33. A. Pietsch. The existence of singular traces on simply generated operator ideals. Integral Equations Operator Theory 92, 1 (2020), Paper No. 7, 23.
  34. A. Pietsch. More about singular traces on simply generated operator ideals. Arch. Math. (Basel) 115, 3 (2020), 299–308.
  35. Fréchet differentiability of the norm of Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-spaces associated with arbitrary von Neumann algebras. C. R. Math. Acad. Sci. Paris 352, 11 (2014), 923–927.
  36. Dixmier measurability in Marcinkiewicz spaces and applications. J. Funct. Anal. 265, 12 (2013), 3053–3066.
  37. Main classes of invariant Banach limits. Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), 124–150.
  38. Segal, I. A non-commutative extension of abstract integration. Ann. of Math. (2) 57 (1953), 401–457.
  39. Banach limits and traces on ℒ1,∞subscriptℒ1\mathcal{L}_{1,\infty}caligraphic_L start_POSTSUBSCRIPT 1 , ∞ end_POSTSUBSCRIPT. Adv. Math. 285 (2015), 568–628.
  40. Sparr, A. On the conjugate space of the Lorentz space L⁢(ϕ,q)𝐿italic-ϕ𝑞L(\phi,q)italic_L ( italic_ϕ , italic_q ). In Interpolation theory and applications, vol. 445 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 313–336.
  41. Dixmier traces and non-commutative analysis. J. Geom. Phys. 105 (2016), 102–122.
  42. Generalized limits with additional invariance properties and their applications to noncommutative geometry. Adv. Math. 239 (2013), 164–189.
  43. Takesaki, M. Theory of operator algebras. I. Springer-Verlag, New York-Heidelberg, 1979.
  44. Takesaki, M. Theory of operator algebras. II, vol. 125 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6.
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free