Completely bounded norms of right module maps

It is well-known that if T is a Dm-Dn bimodule map on the m × 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 D2-module map, then we show that ‖T‖cb=‖T‖. 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 supremumC(m,n)=supT‖cb:Ta rightDn-module map onMm,nwith‖T‖≤1},we show that C(m,m2)=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.