Let m,nβ₯2 be positive integers, Mmβ the set of mΓm complex
matrices and Mnβ the set of nΓn complex matrices. Regard Mmnβ as
the tensor space MmββMnβ. Suppose β£β β£ is the Ky Fan k-norm
with 1β€kβ€mn, or the Schatten p-norm with 1β€pβ€β
(pξ =2) on Mmnβ. It is shown that a linear map Ο:MmnββMmnβ satisfying β£AβBβ£=β£Ο(AβB)β£ for all AβMmβ
and BβMnβ if and only if there are unitary U,VβMmnβ such that
Ο has the form AβBβ¦U(Ο1β(A)βΟ2β(B))V,
where Οiβ(X) is either the identity map Xβ¦X or the
transposition map Xβ¦Xt. The results are extended to tensor space
Mn1βββ...βMnmββ of higher level. The connection of the
problem to quantum information science is mentioned.Comment: 13 page