Non-linear positive maps between $C^*$-algebras

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 ($n\geq 2$). We show that if for a unital $3$-positive map $\Phi: \mathscr{A}\longrightarrow\mathscr{B}$ between unital $C^*$-algebras and some $A\in \mathscr{A}$ equality $\Phi(A^*A)= \Phi(A)^* \Phi(A)$ holds, then $\Phi(XA)=\Phi (X)\Phi (A)$ for all $X \in \mathscr{A}$. In addition, we prove that for a certain class of unital positive maps $\Phi: \mathscr{A}\longrightarrow\mathscr{B}$ between unital $C^*$-algebras, the inequality $\Phi(\alpha A)\leq\alpha \Phi(A)$ holds for all $\alpha \in [0,1]$ and all positive elements $A\in \mathscr{A}$ if and only if $\Phi(0)=0$. Furthermore, we show that if for some $\alpha$ in the unit ball of $\mathbb{C}$ or in $\mathbb{R}_+$ with $|\alpha|\neq 0,1$, the equality $\Phi(\alpha I)=\alpha I$ holds, then $\Phi$ is additive on positive elements of $\mathscr{A}$. Moreover, we present a mild condition for a $6$-positive map, which ensures its linearity.