Elementary operators on Hilbert modules over prime C*-algebras

Abstract 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 ϕ : X → X for which there exists a finite number of elements u i in the C ⁎ -algebra B ( X ) of adjointable operators on X and v i in the multiplier algebra M ( A ) of A such that ϕ ( x ) = ∑ i u i x v i for x ∈ 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 B ( X ) ⊗ M ( A ) if and only if A is a prime C ⁎ -algebra.