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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