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Completely bounded norms of right module maps
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Completely bounded norms of right module maps  
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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.
Journal: Linear Algebra and Its Applications - LINEAR ALGEBRA APPL , 2011
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