Non-linear positive maps between C-algebras

We present some properties of (not necessarily linear) positive maps between -algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between -algebras. Then we give some basic properties and fundamental inequalities related to such maps. Next, we study n-positive maps (...

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Bibliographic Details
Published in:Linear & multilinear algebra 2020-08, Vol.68 (8), p.1501-1517
Main Authors: Dadkhah, Ali, Moslehian, Mohammad Sal
Format: Article
Language:English
Subjects:
Algebra
Lieb map
Linearity
n-positive map
nonlinear map
superadditive
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Summary:We present some properties of (not necessarily linear) positive maps between -algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between -algebras. Then we give some basic properties and fundamental inequalities related to such maps. Next, we study n-positive maps ( ). We show that if for a unital 3-positive map between unital -algebras and some equality holds, then for all . In addition, we prove that for a certain class of unital positive maps between unital -algebras, the inequality holds for all and all positive elements if and only if . Furthermore, we show that if for some α in the unit ball of or in with , the equality holds, then Φ is additive on positive elements of . Moreover, we present a mild condition for a 6-positive map, which ensures its linearity.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2018.1547357