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 supremum C(m,n) = sup{‖T‖cb : T a right Dn-module map on Mm,n with ‖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.