POSITIVE MAP AS DIFFERENCE OF TWO COMPLETELY POSITIVE OR SUPER-POSITIVE MAPS

For a linear map from M-m to M-n, besides the usual positivity, there are two stronger notions, complete positivity and super-positivity. Given a positive linear map phi we study a decomposition phi = phi((1)) - phi((2)) with completely positive linear maps phi((j)) (j = 1,2). Here phi((1)) + phi((2)) is of simple form with norm small as possible. The same problem is discussed with super positivity in place of complete positivity.