Theoretical study reveals structural conditions and algebraic inequalities for positive maps between C*-algebras, highlighting criteria that force non-linear operations into linearity.
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 Φ:A⟶B between unital C∗-algebras and some A∈A equality Φ(A∗A)=Φ(A)∗Φ(A) holds, then Φ(XA)=Φ(X)Φ(A) for all X∈A. In addition, we prove that for a certain class of unital positive maps Φ:A⟶B between unital C∗-algebras, the inequality Φ(αA)≤αΦ(A) holds for all α∈[0,1] and all positive elements A∈A if and only if Φ(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 Φ(αI)=αI holds, then Φ 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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Dadkhah et al. (2018) studied this question.