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Algebra i Analiz, 1998, Volume 10, Issue 1, Pages 132–186 (Mi aa975)  

This article is cited in 42 scientific papers (total in 42 papers)

Research Papers

The similarity degree of an operator algebra

G. Pisierab

a Université Paris VI, Paris, France
b Texas A\&M University, College Station, TX
Abstract: Let A be a unital operator algebra having the property that every bounded unital homomorphism u:AB(H) is similar to a contractive one. Let Sim(u)=inf{SS1}, where the infimum runs over all invertible operators S:HH such that the “conjugate” homomorphism aS1u(a)S is contractive. Now for all c>1, let Φ(c)=supSim(u), where the supremum runs over all unital homomorphism u:AB(H) with u. Then there is \alpha\ge 0 such that for some constant K we have:
\Phi(c)\le Kc^{\alpha},\qquad c>1.
Moreover, the infimum of such \alpha's is an integer (denoted by d(A) and called the similarity degree of A), and (*) is still true for some K when \alpha=d(A). Among the applications of these results, new characterizations are given of proper uniform algebras on one hand, and of nuclear C^*-algebras on the other. Moreover, a characterization of amenable groups is obtained, which answers (at least partially) a question on group representations going back to a 1950 paper of Dixmier.
Keywords: Similarity problem, similarity degree, completely bounded map, operator space, operator algebra, group representation, uniform algebra.
Received: 05.04.1997
Bibliographic databases:
Document Type: Article
Language: English
Citation: G. Pisier, “The similarity degree of an operator algebra”, Algebra i Analiz, 10:1 (1998), 132–186; St. Petersburg Math. J., 10:1 (1999), 103–146
Citation in format AMSBIB
\Bibitem{Pis98}
\by G.~Pisier
\paper The similarity degree of an operator algebra
\jour Algebra i Analiz
\yr 1998
\vol 10
\issue 1
\pages 132--186
\mathnet{http://mi.mathnet.ru/aa975}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1618400}
\zmath{https://zbmath.org/?q=an:0911.47038}
\transl
\jour St. Petersburg Math. J.
\yr 1999
\vol 10
\issue 1
\pages 103--146
Linking options:
  • https://www.mathnet.ru/eng/aa975
  • https://www.mathnet.ru/eng/aa/v10/i1/p132
  • This publication is cited in the following 42 articles:
    1. Gerasimova M., Gruber D., Monod N., Thom A., “Asymptotics of Cheeger Constants and Unitarisability of Groups”, J. Funct. Anal., 278:11 (2020), UNSP 108457  crossref  isi
    2. Clouatre R., Marcoux L.W., “Compact Ideals and Rigidity of Representations For Amenable Operator Algebras”, Studia Math., 244:1 (2019), 25–41  crossref  mathscinet  zmath  isi
    3. Brannan M., Youn S.-G., “On the Similarity Problem For Locally Compact Quantum Groups”, J. Funct. Anal., 276:4 (2019), 1313–1337  crossref  mathscinet  isi  scopus
    4. Lee H.H., Samei E., Spronk N., “Similarity Degree of Fourier Algebras (Vol 271, Pg 593, 2016)”, J. Funct. Anal., 277:3 (2019), 958–964  crossref  isi
    5. Roydor J., “Dual Operator Algebras Close to Injective Von Neumann Algebras”, Pac. J. Math., 293:2 (2018), 407–426  crossref  mathscinet  zmath  isi  scopus
    6. Hadwin D., Qian W., Shen J., “Similarity Degree of Type II1 Von Neumann Algebras With Property Gamma”, J. Operat. Theor., 79:2 (2018), 269–285  crossref  mathscinet  zmath  isi  scopus
    7. Miglioli M., Schlicht P., “Geometric Spects a of Similarity Problems”, Int. Math. Res. Notices, 2018, no. 23, 7171–7197  crossref  mathscinet  isi  scopus
    8. Pisier G., “On a Linearization Trick”, Enseign. Math., 64:3-4 (2018), 315–326  crossref  isi
    9. Pop F., “Similarities of Tensor Products of Type II1 Factors”, Integr. Equ. Oper. Theory, 89:3 (2017), 455–463  crossref  mathscinet  zmath  isi  scopus
    10. Qian W., Shen J., “Similarity Degree of a Class of C^*-Algebras”, Integr. Equ. Oper. Theory, 84:1 (2016), 121–149  crossref  mathscinet  zmath  isi  scopus
    11. Lee H.H., Samei E., Spronk N., “Similarity degree of Fourier algebras”, J. Funct. Anal., 271:3 (2016), 593–609  crossref  mathscinet  zmath  isi  scopus
    12. Marcoux L.W., Popov A.I., “Abelian, amenable operator algebras are similar to C^{*} -algebras”, Duke Math. J., 165:12 (2016), 2391–2406  crossref  mathscinet  zmath  isi  scopus
    13. Qian W.H., Hadwin D., “Universal C^*-Algebras Defined By Completely Bounded Unital Homomorphisms”, Acta. Math. Sin.-English Ser., 31:12 (2015), 1825–1844  crossref  mathscinet  zmath  isi
    14. Wang LiGuang, “on the Properties of Some Sets of Von Neumann Algebras Under Perturbation”, Sci. China-Math., 58:8 (2015), 1707–1714  crossref  mathscinet  zmath  isi  scopus
    15. Cameron J., Christensen E., Sinclair A.M., Smith R.R., White S., Wiggins A.D., “Kadison-Kastler Stable Factors”, Duke Math. J., 163:14 (2014), 2639–2686  crossref  mathscinet  zmath  isi  scopus
    16. Dong Zh., Zhao Ya.F., “A Weak Similarity Degree Characterization For Injective Von Neumann Algebras”, Acta. Math. Sin.-English Ser., 30:10 (2014), 1689–1697  crossref  mathscinet  zmath  isi  scopus
    17. Ricard E., Roydor J., “A Noncommutative Amir-Cambern Theorem For Von Neumann Algebras and Nuclear C^*-Algebras”, J. Funct. Anal., 267:4 (2014), 1121–1136  crossref  mathscinet  zmath  isi  scopus
    18. Dickson L., “A Kadison-Kastler Row Metric and Intermediate Subalgebras”, Int. J. Math., 25:8 (2014), 1450082  crossref  mathscinet  zmath  isi  scopus
    19. Wu J.S., Wu W.M., “Similarity Degrees For the Crossed Product of Von Neumann Algebras”, Acta. Math. Sin.-English Ser., 30:5 (2014), 723–736  crossref  mathscinet  zmath  isi  scopus
    20. Hadwin D., Li W., “The Similarity Degree of Some C^*-Algebras”, Bull. Aust. Math. Soc., 89:1 (2014), 60–69  crossref  mathscinet  zmath  isi  scopus
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