The noncommutative Choquet boundary II: Hyperrigidity

  • William ArvesonEmail author


A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space AB(H) and every sequence of unital completely positive linear maps ϕ 1, ϕ 2,... from B(H) to itself,
$$\mathop {\lim }\limits_{n \to \infty } ||{\phi _n}(g) - g|| = 0,{\forall _g} \in G \Rightarrow \mathop {\lim }\limits_{n \to \infty } {\phi _n}(a) - a|| = 0,{\forall _a} \in A.$$
We show that one can determine whether a given set G of generators is hyperrigid by examining the noncommutative Choquet boundary of the operator space spanned by GG*. We present a variety of concrete applications and discuss prospects for further development.


Irreducible Representation Compact Operator Boundary Representation Separable Hilbert Space Boundary Ideal 
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  1. [Arv69]
    W. Arveson, Subalgebras of C*-algebras, Acta Mathematica 123 (1969), 141–224.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Arv72]
    W. Arveson, Subalgebras of C*-algebras II, Acta Mathematica 128 (1972), 271–308.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Arv08]
    W. Arveson, The noncommutative Choquet boundary, Journal of the American Mathematical Society 21 (2008), 1065–1084. arXiv:OA/0701329v4.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Bau61]
    H. Bauer, Silovscher rand und Dirichletsches problem, Université de Grenoble. Annales de l’Institut Fourier 11 (1961), 89–136.zbMATHGoogle Scholar
  5. [BD78]
    H. Bauer and K. Donner, Korovkin approximation in C 0(X), Mathematische Annalen 236 (1978), 225–237.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [BS55]
    J. Bendat and S. Sherman, Monotone and convex operator functions, Transactions of the American Mathematical Society 79 (1955), 58–71.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Cho74]
    M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras Illinois Journal of Mathematics 18 (1974), 565–574.MathSciNetzbMATHGoogle Scholar
  8. [Dav57]
    C. Davis, A schwarz inequality for convex operator functions, Proceedings of the American Mathematical Society 8 (1957), 42–44.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [DM05]
    M. Dritschel and S. McCullough, Boundary representations for families of representations of operator algebras and spaces, Journal of Operator Theory 53 (2005), 159–167.MathSciNetzbMATHGoogle Scholar
  10. [Don82]
    K. Donner, Extension of positive operators and Korovkin theorems, Lecture Notes in Mathematics 904, Springer-Verlag, Berlin, 1982.zbMATHGoogle Scholar
  11. [Gli67]
    I. Glicksberg, The abstract F. and M. Riesz theorem, Journal of Functional Analysis 1 (1967), 109–122.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [Ham79a]
    M. Hamana, Injective envelopes of C*-algebras, Journal of the Mathematical Society of Japan 31 (1979), 181–197.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Ham79b]
    M. Hamana, Injective envelopes of operator systems Kyoto University. Research Institute for Mathematical Sciences. Publications 15 (1979), 773–785.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [JOR03]
    M. Junge, N. Ozawa, and Z. Ruan, On OL structures of nuclear C*-algebras, Mathematische Annalen 325 (2003), 449–483.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Kor53]
    P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Doklady Akademii Nauk SSSR (N.S.) 90 (1953), 961–964.MathSciNetzbMATHGoogle Scholar
  16. [Kor60]
    P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan publishing Corp., Delhi, 1960.Google Scholar
  17. [NS06]
    S. Neshveyev and E. Størmer, Dynamical Entropy in Operator Algebras, Ergebnisse der mathematik und ihrer grenzgebiete, Springer, Hiedelberg, 2006.zbMATHGoogle Scholar
  18. [Pet86]
    D. Petz, On the equality in Jensen’s inequality for operator convex functions, Integral Equations and Operator Theory 9 (1986), 744–747.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [Phe66]
    R. R. Phelps, Lectures on Choquet’s Theorem, Volume 7 of Van Nostrand Mathematical Studies, Van Nostrand, Princeton, 1966.zbMATHGoogle Scholar
  20. [Phe01]
    R. R. Phelps, Lectures on Choquet’s Theorem, Lecture Notes in Mathematics 1757, second edition, Springer-Verlag, Berlin, 2001.zbMATHCrossRefGoogle Scholar
  21. [Šaš67]
    Yu. A. Šaškin, The Milman-Choquet boundary and approximation theory, Functional Analysis and its Applications 1 (1967), 170–171.Google Scholar

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© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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