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Эта публикация цитируется в 42 научных статьях (всего 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
Образец цитирования:
G. Pisier, “The similarity degree of an operator algebra”, Алгебра и анализ, 10:1 (1998), 132–186; St. Petersburg Math. J., 10:1 (1999), 103–146
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/aa975 https://www.mathnet.ru/rus/aa/v10/i1/p132
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