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*-normsAuthors: 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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