Elementary operators on Hilbert modules over prime C⁎-algebras
Journal of Mathematical Analysis and Applications
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 xv_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
... ators 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.