, Volume 16, Issue 1, pp 53–66 | Cite as

The Kadison–Singer problem for the direct sum of matrix algebras

  • Charles Akemann
  • Joel Anderson
  • Betul Tanbay
Open Access


Let M n denote the algebra of complex n × n matrices and write M for the direct sum of the M n . So a typical element of M has the form
$$ x = x_1\oplus x_2 \cdots \oplus x_n \oplus \cdots, $$
where \({x_n \in M_n}\) and \({\|x\| = \sup_n\|x_n\|}\). We set \({D= \{\{x_n\}\in M: x_n\,{\rm is\,diagonal\,for\,all}\,N\}}\). We conjecture (contra Kadison and Singer in Am J Math 81:383–400, 1959) that every pure state of D extends uniquely to a pure state of M. This is known for the normal pure states of D, and we show that this is true for a (weak*) open, dense subset of all the singular pure states of D. We also show that (assuming the Continuum hypothesis) M has pure states that are not multiplicative on any maximal abelian *-subalgebra of M.


Kadison–Singer problem Matrix algebras Operator algebras 

Mathematics Subject Classification (2000)

Primary 46L 


Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Akemann, C.A., Anderson, J.: Lyapnov theorems for operator algebras. Mem. Am. Math. Soc. 458 (1991)Google Scholar
  2. 2.
    Akemann C.A., Anderson J., Pedersen G.K.: Excising states of C*-algebras. Can. J. Math. 38(5), 223–230 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Akemann C.A., Eilers S.: Regularity of projections revisited. J. Oper. Theory 48, 515–534 (2002)MathSciNetMATHGoogle Scholar
  4. 4.
    Akemann C.A., Weaver N.: Classically normal pure states. Positivity 11, 617–625 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Anderson J.: Extreme points in sets of positive maps on B(H). J. Funct. Anal. 31, 195–217 (1979)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Anderson J.: Extensions, restrictions and representations of states on C *-algebras. Trans. Am. Math. Soc. 249, 303–329 (1979)MATHGoogle Scholar
  7. 7.
    Anderson, J.: Pathology in the Calkin algebra. In: Topics in Modern Operator Theory, pp. 27–43. Birkhaüser, Boston (1981)Google Scholar
  8. 8.
    Dixmier J.: C*-Algebras. North Holland, London (1977)MATHGoogle Scholar
  9. 9.
    Feldman J.: Nonseparability of certain finite factors. Proc. Am. Math. Soc. 7, 23–26 (1956)MATHCrossRefGoogle Scholar
  10. 10.
    Friedman, J.: A Proof of Alon’s second eigenvalue conjecture and related problems. Mem. Am. Soc. 195 (2008)Google Scholar
  11. 11.
    Johnson B.E., Parrott S.K.: Operators commuting with a von Neumann algebra modulo the set of compact operators. J. Funct. Anal. 11, 39–61 (1972)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kadison R.V., Ringrose J.: Fundamentals of the Theory of Operator Algebras, vol. 2. Academic Press, London (1986)Google Scholar
  13. 13.
    Kadison R.V., Singer I.M.: Extensions of pure states. Am. J. Math. 81, 383–400 (1959)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Pedersen G.K.: C*-Algebras and Their Automorphism Groups. Academic Press, London (1979)MATHGoogle Scholar
  15. 15.
    Takesaki M.: Singularity of positive functionals. Proc. Jpn. Acad. 35, 365–366 (1959)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Takesaki M.: Theory of Operator Algebras I. Springer-Verlag, New York (1979)MATHCrossRefGoogle Scholar
  17. 17.
    Tanbay, B.: Extensions of pure states on algebras of operators. Dissertation, University of California at Berkeley (1989)Google Scholar
  18. 18.
    Tanbay B.: Pure state extensions and compressibility of the l 1-algebra. Proc. Am. Soc. 113, 707–713 (1991)MathSciNetMATHGoogle Scholar
  19. 19.
    Tanbay B.: Approximating the averaging operator. Tr. J. Math. 16, 85–94 (1992)MathSciNetMATHGoogle Scholar
  20. 20.
    Wright F.B.: A reduction for algebras of finite type. Ann. Math. (Second Series) 60(3), 560–570 (1954)MATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsPenn State UniversityUniversity ParkUSA
  3. 3.Department of MathematicsBogazici UniversityBebek, IstanbulTurkey

Personalised recommendations