Mathematische Zeitschrift

, Volume 266, Issue 3, pp 693–705 | Cite as

Ideals of a C *-algebra generated by an operator algebra



In this paper, we consider ideals of a C *-algebra \({C^*(\mathcal{B})}\) generated by an operator algebra \({\mathcal{B}}\) . A closed ideal \({J\subseteq C^*(\mathcal{B})}\) is called a K-boundary ideal if the restriction of the quotient map on \({\mathcal{B}}\) has a completely bounded inverse with cb-norm equal to K −1. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from \({C^*(\mathcal{B})}\) onto \({\mathcal{B}}\) which is a projection on \({\mathcal{B}}\) is completely bounded. Moreover, we prove that Kadison’s similarity problem is decided on one particular C *-algebra which is a completion of the *-double of \({M_2(\mathbb{C})}\) .

Mathematics Subject Classification (2000)

46L05 46L07 47L55 47L07 47L30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albeverio S., Juschenko K., Proskurin D., Samoilenko Yu.: *-Wildness of some classes of C *-break algebras. Methods Funct. Anal. Topol. 12(4), 315–325 (2006)Google Scholar
  2. 2.
    Arveson W.: Subalgebras of C *-algebras. Acta Math. 123, 141–224 (1969)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arveson, W.: The noncommutative Choquet boundary. arXiv: math/0701329Google Scholar
  4. 4.
    Blecher D.: Completely bounded characterization of operator algebras. Math. Ann. 303, 227–240 (1969)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Blecher D.: Modules over operator algebras and maximal C *-dilation. J. Funct. Anal. 169(1), 251–288 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blecher D., Paulsen V.: Explicit construction of universal operator algebras and applications to polynomial factorization. Proc. AMS 112(3), 839–850 (1991)MATHMathSciNetGoogle Scholar
  7. 7.
    Blecher D., Ruan Z.-J., Sinclair A.: A characterization of operator algebras. J. Funct. Anal. 89, 188–201 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dritschel M., McCullough S.: Boundary representations for families of representations of operator algebras and spaces. J. Oper. Theory 53(1), 159–167 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    Effros, E., Ruan, Z.-J.: Operator spaces, London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  10. 10.
    Gardner T.: On isomorphisms of C *-algebras. Am. J. Math. 87, 384–396 (1965)MATHCrossRefGoogle Scholar
  11. 11.
    Haagerup U.: Solution of the similarity problem for cyclic representations of C *-algebras. Ann. Math. 118, 215–240 (1983)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hamana M.: Injective envelopes of operator systems. Publ. Res. Inst. Math. Sci. 15, 773–785 (1979)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Johnson B.: The uniqueness of the (complete) norm topology. Bull. Am. Math. Soc. 73, 537–539 (1967)MATHCrossRefGoogle Scholar
  14. 14.
    Juschenko K.: *-Wildness of a semidirect product of F 2 and a finite group. Methods Funct. Anal. Topol. 11(4), 376–382 (2005)Google Scholar
  15. 15.
    Juschenko K., Sukretniy K.: On *-wildness of a free product of finite-dimensional C *-algebras. Methods Funct. Anal. Topol. 12(2), 151–156 (2006)Google Scholar
  16. 16.
    Juschenko K., Popovych S.: Matrix ordered operator algebras. Indiana Univ. J. Math. 58, 1203–1218 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kissin, E., Shulman, V.: Representations on Krein spaces and derivations of C *-algebras. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 89. CRC Press (1997)Google Scholar
  18. 18.
    Ostrovskyi, V., Samoilenko, Yu.: Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators. Reviews in Mathematics and Mathematical Physics, vol. 11, pt.1. Harwood Academic Publishers, Amsterdam (1999)Google Scholar
  19. 19.
    Paulsen V.: Every completely polynomially bounded operator is similar to a contaraction. J. Funct. Anal. 55(1), 1–17 (1984)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Paulsen, V.: Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)Google Scholar
  21. 21.
    Pisier G.: Similarity problems and completely bounded maps. Second, expanded edition. Includes the solution to “The Halmos problem”. J. Am. Math. Soc. 10, 351–369 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pisier G.: The similarity degree of an operator algebra. Algebra i Analiz 10(1), 132–186 (1998)MATHMathSciNetGoogle Scholar
  23. 23.
    Pisier, G.: A polynomially bounded operator on Hilbert space which is not similar to a contraction. Lecture Notes in Mathematics, vol. 1618. Springer, Berlin (2001)Google Scholar
  24. 24.
    Pitts, D.: Norming algebras and automatic complete boundedness of isomorphism of operator algebras. arXiv: math.OA/0609604 (2006)Google Scholar
  25. 25.
    Pop F., Sinclair A., Smith R.: Norming C *-algebras by C *-subalgebras. J. Funct. Anal. 175(1), 168–196 (2000)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Popovych S.: *-Doubles and embedding of associative algebras in \({{\bf B}(\mathcal{H})}\) . Indiana Univ. J. Math. 57, 3443–3462 (2008)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Shulman V.: On representations of C *-algebras on indefinite metric spaces. Mat. Zametki 22, 583–592 (1977)MathSciNetGoogle Scholar
  28. 28.
    Smith R.: Completely bounded maps between C *-algebras. J. London Math. Soc. 27, 157–166 (1983)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Smith R.: Completely bounded maps and the Haagerup tensor product. J. Funct. Anal. 102(1), 156–175 (1991)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of MathematicsChalmers University of TechnologyGöteborgSweden

Personalised recommendations