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 example...

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Bibliographic Details
Published in:Linear algebra and its applications 2012-03, Vol.436 (5), p.1406-1424
Main Authors: Levene, Rupert H., Timoney, Richard M.
Format: Article
Language:English
Subjects:
Algebra
Completely bounded
Exact sciences and technology
Fidelity
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Matrix numerical range
Operator theory
Right module map
Sciences and techniques of general use
Tracial geometric mean
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2011.08.036