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Let $X$ be a right Hilbert module over a $C^*$- algebra $A$ equipped with the canonical operator space structure. We define an elementary operator on $X$ as a map $phi : X to X$ for which there exists a finite number of elements $u_i$ in the $C^*$-algebra $mathbb{; ; ; ; B}; ; ; ; (X)$ of adjointable operators on $X$ and $v_i$ in the multiplier algebra $M(A)$ of $A$ such that $phi(x)=sum_i u_i xv_i$ for $x in X$. If $X=A$ this notion agrees with the standard notion of an elementary operator on $A$. In this paper we extend Mathieu's theorem for elementary operators on prime $C^*$- algebras by showing that the completely bounded norm of each elementary operator on a non-zero Hilbert $A$- module $X$ agrees with the Haagerup norm of its corresponding tensor in $mathbb{; ; ; ; B}; ; ; ; (X)otimes M(A)$ if and only if $A$ is a prime $C^*$- algebra.
$C^*$-algebra ; prime ; Hilbert $C^*$-module ; elementary operator ; completely bounded map
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