Skip to main content
SpringerLink
Log in

Trace inequalities for Rickart \(C^*\)-algebras

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

Rickart \(C^*\)-algebras are unital and satisfy polar decomposition. We proved that if a unital \(C^*\)-algebra \({\mathcal {A}}\) satisfies polar decomposition and admits “good” faithful tracial states then \({\mathcal {A}}\) is a Rickart \(C^*\)-algebra. Via polar decomposition we characterized tracial states among all states on a Rickart \(C^*\)-algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart \(C^*\)-algebra. For a block projection operator and a trace on a Rickart \(C^*\)-algebra we proved a new inequality. As a corollary, we obtain a sharp estimate for a trace of the commutator of any Hermitian element and a projection. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra.

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. Ara, P.: Left and right projections are equivalent in Rickart \(C^*\)-algebras. J. Algebra 120(2), 433–448 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ara, P.: \(K\)-theory for Rickart \(C^*\)-algebras. \(K\)-Theory 5(3), 281–292 (1991)

  3. Ara, P., Goldstein, D.: A solution of the matrix problem for Rickart \(C^*\)-algebras. Math. Nachr. 164, 259–270 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ara, P., Goldstein, D.: Rickart \(C^*\)-algebras are \(\sigma \)-normal. Arch. Math. (Basel) 65(6), 505–510 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ayupov, S.A.: Classification and representation of ordered Jordan algebras (Russian). “Fan”, Tashkent (1986)

  6. Ayupov, Sh.A., Arzikulov, F.N.: Jordan counterparts of Rickart and Baer \(\ast \)-algebras. Uzbek. Mat. Zh. 1, 13–33 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Ayupov, Sh.A., Arzikulov, F.N.: Jordan counterparts of Rickart and Baer \(*\)-algebras, II. São Paulo. J. Math. Sci. 13(1), 27–38 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Berberian, S.K.: Baer *-rings. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  9. Bikchentaev, A.M.: On a property of \(L_p\)-spaces on semifinite von Neumann algebras. Math. Notes 64(1–2), 159–163 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bikchentaev, A.M., Tikhonov, O.E.: Characterization of the trace by monotonicity inequalities. Linear Algebra Appl. 422(1), 274–278 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bikchentaev, A.M.: Commutativity of projections and trace characterization on von Neumann algebras. Sib. Math. J. 51(6), 971–977 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bikchentaev, A.M.: Commutation of projections and trace characterization on von Neumann algebras. II. Math. Notes 89(3–4), 461–471 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bikchentaev, A.M.: The Peierls-Bogoliubov inequality in \(C^*\)-algebras and characterization of tracial functionals. Lobachevskii J. Math. 32(3), 175–179 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bikchentaev, A.M.: A block projection operator in normed ideal spaces of measurable operators. Russ. Math. (Iz. VUZ) 56(2), 75–79 (2012)

  15. Bikchentaev, A.M.: Commutativity of operators and characterization of traces on \(C^*\)-algebras. Dokl. Math. 87(1), 79–82 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bikchentaev, A.M.: Commutation of projections and characterization of traces on von Neumann algebras. III. Int. J. Theor. Phys. 54(12), 4482–4493 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bikchentaev, A.M.: Inequality for a trace on a unital \(C^*\)-algebra. Math. Notes 99(4), 487–491 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bikchentaev, A.M.: On operator monotone and operator convex functions. Russ. Math. (Iz. VUZ) 60(5), 61–65 (2016)

  19. Bikchentaev, A.M.: Differences of idempotents in \(C^*\)-algebras. Sib. Math. J. 58(2), 183–189 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bikchentaev, A.M.: Differences of idempotents in \(C^*\)-algebras and the quantum Hall effect. Theoret. Math. Phys. 195(1), 557–562 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bikchentaev, A.M.: Trace and differences of idempotents in \(C^*\)-algebras. Math. Notes 105(5–6), 641–648 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bikchentaev, A.M.: Metrics on projections of the von Neumann algebra associated with tracial functionals. Sib. Math. J. 60(6), 952–956 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bikchentaev, A.M., Abed, S.A.: Projections and traces on von Neumann algebras. Lobachevskii J. Math. 40(9), 1260–1267 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bikchentaev, A.M.: Inequalities for determinants and characterization of the trace. Sib. Math. J. 61(2), 248–254 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bikchentaev, A.: Sukochev, F.: Inequalities for the block projection operators. J. Funct. Anal. 280(7), article 108851 (2021)

  26. Blackadar, B.E., Cuntz, J.: The structure of stable algebraically simple \(C^\ast \)-algebras. Am. J. Math. 104(4), 813–822 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  27. Blackadar, B., Handelman, D.: Dimension functions and traces on \(C^\ast \)-algebras. J. Funct. Anal. 45(3), 297–340 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Blackadar, B.: Projections in \(C^\ast \)-algebras, in: R.S. Doran (Ed.), \(C^*\)-Algebras: 1943–1993, A Fifty Year Celebration, in: Contemporary Mathematics, vol. 167, pp. 115–129, Amer. Math. Soc., Providence, R.I. (1994)

  29. Blackadar, B.: Operator algebras, theory of \(C^*\)-algebras and von Neumann algebras, in: Encyclopaedia of Mathematical Sciences, in: Operator Algebras and Non-commutative Geometry, III, vol. 122, Springer-Verlag, Berlin (2006)

  30. Busqué, C.: \(K\)-theoretic triviality for Rickart \(C^\ast \)-algebras and \(\aleph _0\)-continuous regular rings. Proc. Am. Math. Soc. 101(1), 24–28 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Chilin, V., Krygin, A., Sukochev, F.: Extreme points of convex fully symmetric sets of measurable operators. Integr. Equ. Oper. Theory 15(2), 186–226 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Cho, K., Sano, T.: Young’s inequality and trace. Linear Algebra Appl. 431(8), 1218–1222 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer Study Edition. Springer, Berlin (1987)

    MATH  Google Scholar 

  34. Dixmier, J.: \(C^{\ast } \)-algebras and their representations (Russian). Translated from the French. “Nauka”, Moscow (1974)

  35. Elliott, G.A., Robert, L., Santiago, L.: The cone of lower semicontinuous traces on a \(C^*\)-algebra. Am. J. Math. 133(4), 969–1005 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gardner, L.T.: An inequality characterizes the trace. Can. J. Math. 31(6), 1322–1328 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  37. Gohberg, I.C., Kreǐn, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. Mathem. Monographs, Vol. 18, Amer. Math. Soc., Providence, R.I. (1969)

  38. Goldstein, D.: Polar decomposition in Rickart \(C^*\)-algebras. Publ. Mat. 39(1), 5–21 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. Goldstein, DSh., Katz, A.A.: On the algebra of measurable operators affiliated to a Rickart ordered \(\ast \)-algebra. Far East J. Math. Sci. (FJMS) 24(1), 45–54 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Goldstein, D.Sh., Katz, A.A.: On a local structure in Rickart algebras—definitions and basic properties. Int. J. Funct. Anal. Oper. Theory Appl. 2(2), 133–168 (2010)

  41. Haagerup, U.: Quasitraces on exact \(C^*\)-algebras are traces. C. R. Math. Acad. Sci. Soc. R. Can. 36(2–3), 67–92 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Halmos, P.R., Sunder, V.S.: Bounded integral operators on \(L^{2}\) spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 96. Springer Berlin-New York (1978)

  43. Hoa, D.T., Osaka, H., Toan, H.M.: On generalized Powers-Størmer’s inequality. Linear Algebra Appl. 438(1), 242–249 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras. Vol. I. Elementary theory. Graduate Studies in Mathematics, 15. Amer. Math. Soc., Providence, R.I. (1997)

  45. Kaplansky, I.: Projections in Banach algebras. Ann. Math. 53(2), 235–249 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lord, S., Sukochev, F., Zanin, D.: Singular Traces. Theory and Applications, de Gruyter Studies in Mathematics, Vol. 46 (2013)

  47. Menal, P., Moncasi, J.: Lifting units in self-injective rings and an index theory for Rickart \(C^\ast \)-algebras. Pac. J. Math. 126(2), 295–329 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  48. Petz, D., Zemánek, J.: Characterizations of the trace. Linear Algebra Appl. 111, 43–52 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  49. Rickart, C.E.: Banach algebras with an adjoint operation. Ann. Math. 47(3), 528–550 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  50. Robert, L.: On the comparison of positive elements of a \(C^*\)-algebra by lower semicontinuous traces. Indiana Univ. Math. J. 58(6), 2509–2515 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Rørdam, M.: Fixed-points in the cone of traces on a \(C^{\ast }\)-algebra. Trans. Am. Math. Soc. 371(12), 8879–8906 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  52. Saitô, K., Wright, J.D.M.: On defining \({\rm AW}^*\)-algebras and Rickart \({\rm C}^*\)-algebras. Q. J. Math. 66(3), 979–989 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sherstnev, A.N.: Methods of bilinear forms in non-commutative measure and integral theory. Russian), Fizmatlit, Moscow (2008)

    MATH  Google Scholar 

  54. Stolyarov, A.I., Tikhonov, O.E., Sherstnev, A.N.: Characterization of normal traces on von Neumann algebras by inequalities for the modulus. Math. Notes 72(3–4), 411–416 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  55. Takesaki, M.: Theory of operator algebras. I. Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin (2002)

  56. Takesaki, M.: Theory of Operator Algebras, Operator Algebras and Non-Commutative Geometry, 6. v. II, Encyclopaedia Math. Sci., 125, Springer-Verlag, New York (2003)

  57. Tanaka, R.: \(C^*\)-algebras with weak\(^*\) angelic dual balls. Proc. Am. Math. Soc. Ser. B 7, 127–137 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tikhonov, O.E.: Subadditivity inequalities in von Neumann algebras and characterization of tracial functionals. Positivity 9(2), 259–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  59. Upmeier, H.: Automorphism groups of Jordan \(C^* \)-algebras. Math. Z. 176(1), 21–34 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Research was supported by the development program of the Scientific and Educational Mathematical Center of the Volga Federal District (075-02-2020-1478).

Author information

Authors and Affiliations

  1. Kazan Federal University, 18 Kremlyovskaya str., Kazan, Russia, 420008

    Airat Bikchentaev

Authors
  1. Airat Bikchentaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Airat Bikchentaev.

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

Bikchentaev, A. Trace inequalities for Rickart \(C^*\)-algebras. Positivity 25, 1943–1957 (2021). https://doi.org/10.1007/s11117-021-00852-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-021-00852-3

Keywords

Mathematics Subject Classification

Search

Navigation