OPERATOR ALGEBRAS AND QUANTUM GROUPS BANACH CENTER PUBLICATIONS, VOLUME 98 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2012 POSITIVE LINEAR MAPS OF MATRIX ALGEBRAS W. A. MAJEWSKI Institute of Theoretical Physics and Astrophysics, Gda´ nsk University Wita Stwosza 57, 80-952 Gda´ nsk, Poland E-mail: fizwam@univ.gda.pl Dedicated to Professor S. L. Woronowicz, on the occasion of his 70-th birthday Abstract. A characterization of the structure of positive maps is presented. This sheds some more light on the old open problem studied both in Quantum Information and Operator Alge- bras. Our arguments are based on the concept of exposed points, links between tensor products and mapping spaces and convex analysis. 1. Introduction. Physicists are accustomed to pass freely from the Heisenberg picture to the Schr¨ odinger picture and vice versa. As it is well known, the Heisenberg picture deals with observables while the Schr¨ odinger picture concentrates on states, and both pictures are fitted in a dual pair. Time evolutions provided by the pictures are equivalent, and to formulate laws of dynamics within the Heisenberg picture which are compatible with the duality, dynamical maps should be described by positive, continuous, unital maps. Consequently, the concept of continuous, unital, positive maps is at the heart of mathematical foundations of Quantum Theory, and therefore a characterization of the structure of this set is of paramount importance for both Quantum Mechanics and Quan- tum Information. The important point to note here is our assumption on continuity and normalization of positive maps. Both assumptions are indispensable when one studies dynamical maps (in Quantum Mechanics) or in an analysis of states (in Quantum Infor- mations where one considers a composition of a given state with a positive map). The intention of our lecture is twofold. First, we give a brief exposition of our recent results on the structure of the set of positive maps. Thus we will frequently quote our results from [15]. Second, we want to indicate that the given characterization is working nicely, i.e. to show by examples how the general theory can be applied to very concrete models. In that way we will get a better understanding of the dramatic difference between 2010 Mathematics Subject Classification : Primary 47B60; Secondary 81Q99, 46M05. Key words and phrases : positive maps, projective tensor product, exposed maps. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc98-0-12 [293] c  Instytut Matematyczny PAN, 2012