A simple proof of a theorem of Kirchberg and related results on $C^{\ast }$-norms

Abstract

Recently, E.\ Kirchberg [K1--2] revived the study of pairs of $C^{\ast }$-algebras $A,B$ such that there is only one $C^{\ast }$-norm on the algebraic tensor product $A\otimes B$, or equivalently such that $A{\otimes }_{\min }B=A{\otimes }_{\max }B$. Recall that a $C^{\ast }$-algebra is called nuclear cf.\ [L, EL] if this happens for any $C^{\ast }$-algebra $B$. Kirchberg [K1] constructed the first example of a non-nuclear $C^{\ast }$-algebra such that $A{\otimes }_{\min }{A}^{op}=A{\otimes }_{\max }{A}^{op}$. He also proved the following striking result [K2] for which we give a very simple proof and which we extend. \proclaim Theorem 0.1. {\bf (Kirchberg [K2]).} Let $F$ be any free group and let $C^{\ast }(F)$ be the (full) $C^{\ast }$-algebra of $F$, then $$C^{\ast }(F){\otimes }_{\min }B(H)=C^{\ast }(F){\otimes }_{\max }B(H).$$

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