Lieb functions and m-positivity of norms

Non-linear positive maps between C*-algebras

“…Note that [14,Theorem 6.3.9] states that the Hadamard power [a p ij ] of a positive semidefinite 3 × 3 matrix [a ij ] with nonnegative entries is positive semidefinite for p ≥ 1. Hence, according to [13,Section 7.5 Problem 4], positive semidefiniteness of a 3 × 3 matrix [a ij ] implies positive semidefiniteness of [|a i,j |], which ensures Φ p (1 < p < 2) is 3-positive; (ii) every positive semidefinite matrix P ∈ M n (C) induces a map φ P : M n (C) → C defined by φ P (A) = |tr(AP )| such that it is a 3-positive semi-norm on M n (C) (see [10,Lemma 6.1]). We show that φ P is in the class S…”

“…Günther and Klotz [10] investigated n-positive norms on M k (C) as n-positive maps. They show that there is not any 4-positive norm on M k (C).…”

“…. However, φ P can not be 4-positive (see[10, Corollary 2.3]); (iii) It is easy to check that every positive linear functional ϕ : A −→ C on a C * -algebra induces a non-linear 3-positive map Φ : A −→ C that belongs to the class S(2) mon+ given by Φ(A) = |ϕ(A)|, since every positive linear functional is completely positive and the map | · | on C is 3-positive and in the class S (2) mon+ . We now show that if Φ : A → B is in the class S (2) mon+ , then it is strongly superadditive on positive elements of A .…”

Continuity of positive nonlinear maps between $C^*$-algebras

“…He also gave some relationships between the n-positivity, n-convexity, and differentiability of real functions. Recently, some characterizations of power functions that preserve positivity, monotonicity, and convexity were given by Guillot et al [10,11], and the class of n-positive norms on M n was investigated by Günther and Klotz [12]. The authors of [8] extended some properties of positive linear maps to general positive nonlinear maps between C * -algebras.…”

“…For other examples of n-positive and n-monotone nonlinear maps, we refer the reader to [8,12,14,15,20].…”