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

This article is cited in 40 scientific papers (total in 40 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 $u\colon A\to B(H)$ is similar to a contractive one. Let $\operatorname{Sim}(u)=\inf\{\|S\| \|S^{-1}\|\}$, where the infimum runs over all invertible operators $S\colon H\to H$ such that the “conjugate” homomorphism $a\mapsto S^{-1}u(a)S$ is contractive. Now for all $c>1$, let $\Phi(c)=\sup\operatorname{Sim}(u)$, where the supremum runs over all unital homomorphism $u\colon A\to B(H)$ with $\|u\|\le c$. 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.

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English version:
St. Petersburg Mathematical Journal, 1999, 10:1, 103–146

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Received: 05.04.1997

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
\by G.~Pisier
\paper The similarity degree of an operator algebra
\jour Algebra i Analiz
\yr 1998
\vol 10
\issue 1
\pages 132--186
\jour St. Petersburg Math. J.
\yr 1999
\vol 10
\issue 1
\pages 103--146

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    22. 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
    23. 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
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    27. 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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    30. 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
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    32. 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
    33. 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
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