Non-linear positive maps between $C^{\ast }$-algebras

Abstract

We present some properties of (not necessarily linear) positive maps between $C^{\ast }$-algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between $C^{\ast }$-algebras. Then we give some basic properties and fundamental inequalities related to such maps. Next, we study $n$-positive maps ($n\ge 2$). We show that if for a unital $3$-positive map $\Phi :\mathcal{A}⟶\mathcal{B}$ between unital $C^{\ast }$-algebras and some $A\in \mathcal{A}$ equality $\Phi (A^{\ast }A)=\Phi (A{)}^{\ast }\Phi (A)$ holds, then $\Phi (XA)=\Phi (X)\Phi (A)$ for all $X\in \mathcal{A}$. In addition, we prove that for a certain class of unital positive maps $\Phi :\mathcal{A}⟶\mathcal{B}$ between unital $C^{\ast }$-algebras, the inequality $\Phi (\alpha A)\le \alpha \Phi (A)$ holds for all $\alpha \in [0,1]$ and all positive elements $A\in \mathcal{A}$ 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 $|\alpha |\ne 0,1$, the equality $\Phi (\alpha I)=\alpha I$ holds, then $\Phi$ is additive on positive elements of $\mathcal{A}$. Moreover, we present a mild condition for a $6$-positive map, which ensures its linearity.

20 pages, to appear in Linear Multilinear Algebra

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