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ЖУРНАЛЫ // Алгебра и анализ // Архив

Алгебра и анализ, 1998, том 10, выпуск 1, страницы 132–186 (Mi aa975)

Эта публикация цитируется в 42 статьях

Статьи

The similarity degree of an operator algebra

G. Pisierab

a Université Paris VI, Paris, France
b Texas A\&M University, College Station, TX

Аннотация: 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.

Ключевые слова: Similarity problem, similarity degree, completely bounded map, operator space, operator algebra, group representation, uniform algebra.

Поступила в редакцию: 05.04.1997

Язык публикации: английский


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

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