Advertisement

Positivity

pp 1–11 | Cite as

A property of conditional expectation

  • Turdebek N. Bekjan
  • Bolathan K. Sageman
Article
  • 62 Downloads

Abstract

Let \(({\mathcal {M}},\tau )\) be a semi-finite von Neumann algebra, \(({\mathcal {N}},\tau |_{{\mathcal {N}}})\) be a semi-finite von Neumann subalgebra and \({\mathcal {E}}:\;{\mathcal {M}}\rightarrow {\mathcal {N}}\) be a conditional expectation which leaves \(\tau \) invariant. We proved super-majorization for the conditional expectation \({\mathcal {E}}\) and related inequalities.

Keywords

Conditional expectation Super-majorization Semifinite von Neumann algebra 

Mathematics Subject Classification

46L52 47L05 

Notes

Acknowledgements

We thank F. A. Sukochev for suggesting (pointing out) Remark 1. We also thank the referee for very useful comments, which improved the paper. The authors are partially supported by NSFC Grant No. 11771372.

References

  1. 1.
    Antezana, J., Massey, P., Stojanoff, D.: Jensen’s inequality for spectral order and submajorization. J. Math. Anal. Appl. 331, 297–307 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arveson, W.B.: Analyticity in operator algebras. Am. J. Math. 89, 578–642 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bekjan, T.N., Raikhan, M.: An Hadamard-type inequality. Linear Algebra Appl. 443, 228–234 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhatia, R., Choi, M.D., Davis, C.: Comparing a matrix to its off-diagonal part. Oper. Theory Adv. Appl. 40, 151–163 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bekjan, T.N., Ospanov, K.N., Zulkhazhav, A.: Choi-Davis-Jensen inequalities in semifinite von Neumann algebras. J. Funct. Spaces 2015, 1–5 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blecher, D.P., Labuschagne, L.E.: Applications of Fuglede–Kadison determinant: Szegö’s theorem and outers for noncommutative \(H^{p}\). Trans. Am. Math. Soc. 360, 6131–6147 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bourin, J.-C., Hiai, F.: Anti-norms on finite von Neumann algebras. Publ. Res. Inst. Math. Sci. 52(2), 207–235 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brown, L.G., Kosaki, H.: Jensens inequality in semi-finite von Neumann algebras. J. Oper. Theory 23, 3–9 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dirksen, S., de Pagter, B., Potapov, D., Sukochev, F.: Rosenthal inequalities in noncommutative symmetric spaces. J. Funct. Anal. 261, 2890–2925 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dodds, P.G., Dodds, T.K., de Pagter, B.: Fully symmetric operator spaces. Integr. Equat. Oper. Theory 15, 942–972 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dodds, P.G., Dodds, T.K., de Pagter, B.: Non-commutative Köthe duality. Trans. Am. Math. Soc. 339, 717–750 (1993)zbMATHGoogle Scholar
  12. 12.
    Dodds, P.G., Sukochev, F.A.: Submajorisation inequalities for convex and concave functions of sums of measurable operators. Positivity 13(1), 107–124 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fack, T.: Sur la notion de valeur charactéristique. J. Oper. Theory 7, 307–333 (1982)zbMATHGoogle Scholar
  14. 14.
    Fack, T.: Proof of the conjecture of A. Grothendieck on the Fuglede–Kadison determinant. J. Funct. Anal. 50, 215–228 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fack, T., Kosaki, H.: Generalized \(s\)-numbers of \( \tau \)-measurable operators. Pac. J. Math. 123, 269–300 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fuglede, B., Kadison, R.V.: Determinant theory in finite factors. Ann. Math. 55, 520–530 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guido, D., Isola, T.: Singular traces on semifinite von Neumann algebras. J. Funct. Anal. 134, 451–485 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Junge, M., Xu, Q.: Noncommutative maximal ergodic theorems. J. Am. Math. Soc. 20, 385C439 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kalton, N.J., Sukochev, F.A.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lord, S., Sukochev, F.A., Zanin, D.: Singular traces, theory and applications. In: De Gruyter Studies in Mathematics, 46 Google Scholar
  21. 21.
    Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  22. 22.
    Pisier, G., Xu, Q.: Noncommutative \(L^p\)-spaces. Handb. Geom. Banach Spaces 2, 1459–1517 (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Sukochev, F.: Hölder inequality for symmetric operator spaces and trace property of K-cycles. Bull. Lond. Math. Soc. 48(4), 637–647 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sukochev, F.: Completeness of quasi-normed symmetric operator spaces. Indag. Math. (N.S.) 25(2), 376–388 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sedaev, A.A., Semenov, E.M., Sukochev, F.A.: Fully symmetric function spaces without an equivalent Fatou norm. Positivity 19, 419C437 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  27. 27.
    Umegaki, H.: Conditional expectation in an operator algebra. Tôhoku Math. J. 6(2), 177–181 (1954)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiChina

Personalised recommendations