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 Algebra i Analiz: Year: Volume: Issue: Page: Find

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

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

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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://www.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 

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