Completely bounded norms of right module maps

It is well-known that if T is a $D_m-D_n$ bimodule map on the m × n complex matrices, then T is a Schur multiplier and $6T{6}_{\mathit{cb}}=6T6$. If n = 2 and T is merely assumed to be a right D₂-module map, then we show that $6T{6}_{\mathit{cb}}=6T6$. However, this property fails if m ? 2 and n ? 3. For m ? 2 and n = 3, 4 or n ? m² we give examples of maps T attaining the supremum$C(m\text{,}n)=\sup T{6}_{\mathit{cb}}:T\text{a right}{D}_n\text{-module map on}{M}_{m\text{,}n}\text{with}6T6\le 1\}\text{,}$we show that $C(m\text{,}{m}^2)=\sqrt{m}$ and succeed in finding sharp results for C(m, n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on K(H) which is not completely bounded.