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 by n complex matrices, then T is a Schur multiplier and $\|T\|_{cb}=\|T\|$. If n=2 and T is merely assumed to be a right D_2-module map, then we show that $\|T\|_{cb}=\|T\|$. However, this property fails if m>1 and n>2. For m>1 and n=3,4 or $n\geq m^2$, we give examples of maps T attaining the supremum C(m,n)=\sup \|T\|_{cb} taken over the contractive, right D_n-module maps on M_{m,n}, we show that C(m,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 the compact operators K(H) which is not completely bounded.