This paper concerns positive maps between
C-algebras,
particularly when those positive maps are multilinear. We
construct examples of positive bilinear maps that are not
-positive,
and therefore are not completely positive bilinear maps. Paulsen and Smith showed
that completely bounded bilinear maps are in one-to-one correspondence with
completely bounded linear maps. We show that a similar correspondence does not
hold for positive bilinear and linear maps. In particular, we observe that a similar
correspondence does not hold if we replace completely bounded with completely
positive.
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