Elementary operators on Hilbert modules over prime \(C^\ast\)-algebras (Q2304310)

Let \(X\) be a right Hilbert module over a \(C^*\)-algebra \(A\) equipped with the canonical operator space structure and \(\mathbb{B}(X)\) be the set of all adjointable operators on \(X\). A mapping \(\phi : X\to X\) is called an elementary operator if there exist a finite number of elements \(u_i\) in \(\mathbb{B}(X)\) and \(v_i\) in the multiplier algebra \(M(A)\) of \(A\) such that \(\phi (x)=\sum _iu_ixv_i\) for all \(x\in X\). The aim of this paper is to extend \textit{P. Ara} and \textit{M. Mathieu}'s result on elementary operators on prime \(C^*\)-algebras [Local multipliers of \(C^*\)-algebras. London: Springer (2003; Zbl 1015.46001)] to elementary operators on Hilbert modules 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.