Elementary operators on Hilbert modules over prime $C^{\ast }$-algebras

Abstract

Let $X$ be a right Hilbert module over a $C^{\ast }$-algebra $A$ equipped with the canonical operator space structure. We define an elementary operator on $X$ as a map $\varphi :X\to X$ for which there exists a finite number of elements $u_i$ in the $C^{\ast }$-algebra $\mathbb{B}(X)$ of adjointable operators on $X$ and $v_i$ in the multiplier algebra $M(A)$ of $A$ such that $\varphi (x)={\sum }_i{u}_{i}x{v}_i$ 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 $C^{\ast }$-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 $C^{\ast }$-algebra.

10 pages, to appear in J. Math. Anal. Appl

References in corpus (1)

• The cb-norm approximation of generalized skew derivations by elementary operators