Completely Contractive Maps between C∗-algebras

We said that φ is n-positive if φn is positive and that φ is completely positive if φn is positive for all n. The map φ is said to be n-bounded (resp., n-contractive) if ‖φn‖ ≤ c (resp., ‖φn‖ ≤ 1). The map φ is said to be completely bounded (resp., completely contractive) if ‖φ‖cb = supn‖φ‖n <∞ (resp., ‖φ‖cc = supn‖φn‖ ≤ 1). npositivity (resp., n-boundedness or n-contractivity) implies (n−1)-positivity (resp., (n−1)-boundedness or (n−1)-contractivity). The converse is not true in general. For any C∗-algebra A, Mn(Mp(A)) is identified with Mp(Mn(A)) because there is a canonical isomorphism betweenMn(Mp(A)) andMp(Mn(A)) by the rearrangement of ann×nmatrix ofp×p blocks as ap×pmatrix ofn×n blocks with the (i,j)th entry of the (k, )-block becoming the (k, )th entry of the (i,j)th block. This rearrangement corresponds to a preand post-multiplying of a given matrix by a unitary and its adjoint.