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This article is cited in 42 scientific papers (total in 42 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:A→B(H) is similar to a contractive one. Let Sim(u)=inf{‖S‖‖S−1‖}, where the infimum runs over all invertible operators S:H→H
such that the “conjugate” homomorphism a↦S−1u(a)S is contractive. Now for all
c>1, let Φ(c)=supSim(u), where the supremum runs over all unital homomorphism
u:A→B(H) with ‖u‖⩽. 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.
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
Linking options:
https://www.mathnet.ru/eng/aa975 https://www.mathnet.ru/eng/aa/v10/i1/p132
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