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 kTkcb = kTk. If n = 2 and T is merely assumed to be a right D2-module map, then we show that kTkcb = kTk. 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, n) = sup{kTkcb : T a right Dn-module map on Mm,n with kTk ≤ 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.