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

Keywords: C*-algebra, unicity of C*-norms, minimal and maximal tensor product.

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