We present some properties of (not necessarily linear) positive maps between C∗-algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between C∗-algebras. Then we give some basic properties and fundamental inequalities related to such maps. Next, we study n-positive maps (n ≥ 2). We show that if for a unital 3-positive map : A −→ B between unital C∗-algebras and some A ∈ A equality (A∗A) = (A) ∗ (A) holds, then (XA) = (X)(A) for all X ∈ A . In addition, we prove that for a certain class of unital positive maps : A −→ B between unital C∗-algebras, the inequality (αA) ≤ α(A) holds for all α ∈ [0, 1] and all positive elements A ∈ A if and only if (0) = 0. Furthermore, we show that if for some α in the unit ball of C or in R+ with |α| = 0, 1, the equality (αI) = αI holds, then is additive on positive elements of A . Moreover, we present a mild condition for a 6-positive map, which ensures its linearity.

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