Journal of Operator Theory

Volume 35, Issue 2, Spring 1996 pp. 317-335.

A simple proof of a theorem of Kirchberg and related results on C*-norms

Authors: Gilles Pisier
Author institution: Texas A \& M University, College Station, TX 77843, U.S.A. and Universite Paris VI, Equipe d'Analyse, Case 186, 75252 Paris Cedex 05, FRANCE

Summary: Let F be a free group and let C*(F) be the (full) C*-algebra of F. We give a simple proof of Kirchbergâ€™s theorem that there is only one C*-norm on the algebraic tensor product C*(F) \otimes B(H), or equivalently that C*(F) \otimes_min B(H) = C*(F) \otimes_max B(H). More generally, let A be the (unital) free product of a family

$(A_i)_{i \in I}$ of (unital) C*-algebras. We show that if

$A_i \otimes _{{\rm{min}}} B(H) = A_i \otimes _{{\rm{max}}} B(H)$ holds for all i in I, then

$A \otimes _{{\rm{min}}} B(H) = A \otimes _{{\rm{max}}} B(H)$ .

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