The aim of this talk is to present some basic properties of (not necessarily linear) positive maps between C-algebras. We study some classes of n-positive maps. We show that for every unital 3-positive map : A ! B between C-algebras the validity of (AA) = (A)(A) for some A 2 A implies that (XA) = (X)(A) for all X 2 A. Moreover, we show that this result does not hold for non-linear 2-positive maps, in general. We then prove that every 3-positive map : A ! B is superadditive on positive elements whenever (0) = 0, and yield some new results on the n-positivity of any C-norm. Finally, we introduce a class of positive maps : A ! B, and show that the inequality (A) (A) holds for all 2 [0; 1] and all positive elements A 2 A if and only if (0) = 0. Furthermore, we show that if for some in either the unit ball of the complex plane or the positive real numbers with jj 6= 0; 1, the equality (I) = I holds, then is additive on positive elements of A.

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