On the complete boundedness of the Schur block product

We give a Stinespring representation of the Schur block product, say (), on pairs of square matrices with entries in a C-algebra as a completely bounded bilinear operator of the form: A:=(a_{ij}), B:= (b_{ij}): A () B := (a_{ij}b_{ij}) = V pi(A) F pi(B) V, such that V is an isometry, pi is a -representation and F is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new ones on diagonals of matrices. ||A () B|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(AA) \leq A () A \leq diag(AA), and for all vectors f, g: |<A()B f,g> |^2 \leq < diag(AA) g, g> <diag(B*B) f,f> .