Skip to main content
SpringerLink
Log in

Injective Strong Commutativity Preservers on \({\mathcal{T}_{\infty}(F)}\)

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We describe strong commutativity preservers on the algebra of infinite upper triangular matrices over a field F such that char\({(F) \neq 2}\). We show that every such map is some kind of a sum of a few sorts of maps. We also discuss the form of the maps that preserve commutativity in both directions.

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. Baranovsky V.: The variety of pairs of commuting nilpotent matrices is irreducible. Transform. Gropus, 6, 3–8 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basili R.: On the irreducibility of commuting varieties of nilpotent matrices J. Algebra, 268, 58–80 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beasley L.B.: Linear transformations on matrices: The invariance of commuting pairs of matrices. Linear Multilinear Algebra, 6, 179–183 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beasley L.B., Song S.Z.: Linear operators that preserve commuting pairs of nonnegative real matrices. Linear Multilinear Algebra, 51, 79–283 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Bell H.E., Daif M.N.: On commutativity and strong commutativity-preserving maps. Canad. Math. Bull. 37, 443–447 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brešar M., Miers C.R.: Commutativity preserving mappings of von Neumann algebras. Can. J. Math., 45, 695–708 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao Y., Chen Z., Huang C.: Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix spaces. Linear Algebra Appl. 350, 41–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi M.D., Jafarian A.A., Radjavi H.: Linear maps preserving commutativity. Linear Algebra Appl. 87, 227–241 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolinar G., Šemrl P.: Maps on matrix algebras preserving commutativity. Linear Multilinear Algebra 52, 69–78 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guterman A., Li C.K., Šemrl P.: Some general techniques on linear preserver problems. Linear Algebra Appl. 315, 61–81 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Kraft, Geometric methods in representation theory, in: Representations of Algebras, Lecture Notes in Math. 944, Springer-Verlag (New York, 1980), pp. 180–257.

  12. Kunicki C.: Commutativity-preserving and normal-preserving linear transformations on \({\mathcal{M}_{2}}\). Linear Multilinear Algebra, 45, 341–347 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lee T.K., Wong T.L.: Nonadditive strong commutativity preserving maps. Comm. Algebra, 40, 2213–2218 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu C.K., Liau P.K.: Strong commutativity preserving generalized derivations on Lie ideals. Linear Multilinear Algebra 59, 905–915 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Marcoux L.W., Sourour A.R.: Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras. Linear Algebra Appl. 288, 89–104 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miers C.R.: Commutativity preserving maps of factors. Canad. J. Math., 40, 248–256 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Molnár L., Šxemrl P.: Some linear preserver problems on upper triangular matrices. Linear Multilinear Algebra 45, 189–206 (1999)

    Article  MathSciNet  Google Scholar 

  18. Oblak P.: Jordan forms for mutually anihilating nilpotent pairs. Linear Algebra Appl. 428, 1476–1491 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Omladič M.: On operators preserving commutativity. J. Funct. Anal. 66, 105–122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Omladič M., Radjavi H., Šemrl P.: Preserving commutativity. J. Pure Appl. Algebra 156, 309–328 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Petek T.: Additive mappings preserving commutativity. Linear Multilinear Algebra 42, 205–211 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pierce S., Watkins W.: Invariants of linear maps on matrix algebras. Linear Multilinear Algebra 6, 185–200 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schröer J.: Varieties of pairs of nilpotent matrices anihilating each other. Comment. Math. Helv. 79, 396–426 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Šemrl P.: Non-linear commutativity preserving maps. Acta Sci. Math. (Szeged) 71, 781–819 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Słowik R.: Maps on infinite triangular matrices preserving idempotents. Linear Multilinear Algebra 62, 938–964 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Watkins W.: Linear maps that preserve commuting pairs of matrices. Linear Algebra Appl. 14, 29–35 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Watkins, Polynomial functions that preserve commuting pairs of matrices, Linear Multilinear Algebra, 5 (1977/78), 87–90

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100, Gliwice, Poland

    R. Słowik

Authors
  1. R. Słowik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Słowik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Słowik, R. Injective Strong Commutativity Preservers on \({\mathcal{T}_{\infty}(F)}\) . Acta Math. Hungar. 148, 386–404 (2016). https://doi.org/10.1007/s10474-015-0570-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-015-0570-1

Keywords and phrases

Mathematics Subject Classification

Search

Navigation