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## Über dieses Buch

This volume, setting out the theory of positive maps as it stands today, reflects the rapid growth in this area of mathematics since it was recognized in the 1990s that these applications of C*-algebras are crucial to the study of entanglement in quantum theory. The author, a leading authority on the subject, sets out numerous results previously unpublished in book form. In addition to outlining the properties and structures of positive linear maps of operator algebras into the bounded operators on a Hilbert space, he guides readers through proofs of the Stinespring theorem and its applications to inequalities for positive maps.

The text examines the maps’ positivity properties, as well as their associated linear functionals together with their density operators. It features special sections on extremal positive maps and Choi matrices. In sum, this is a vital publication that covers a full spectrum of matters relating to positive linear maps, of which a large proportion is relevant and applicable to today’s quantum information theory. The latter sections of the book present the material in finite dimensions, while the text as a whole appeals to a wider and more general readership by keeping the mathematics as elementary as possible throughout.

## Inhaltsverzeichnis

### Chapter 1. Generalities for Positive Maps

Abstract
In this chapter we introduce the basic concepts concerning positive maps between C -algebras. These maps are linear and carry positive operators to positive operators. We consider several kinds of positive maps, in particular the important completely positive ones, which appear in many applications both in physics and operator algebras. The main part of the paper is devoted to those maps; we show the famous Stinespring theorem, which is the analogue for completely positive maps of the GNS-construction for states. As a consequence we obtain several Schwarz-type inequalities for positive maps. In the last section we consider the adjoint map of a positive map between the bounded operators on a Hilbert space.
Erling Størmer

### Chapter 2. Jordan Algebras and Projection Maps

Abstract
It has been known since the pioneering work of Kadison in the early 1950s that the theory of positive maps is closely related to the Jordan algebra structure of operator algebras. In the first part of the chapter we elaborate on this structure for positive maps. One class of maps for which Jordan algebras are central, are the idempotent ones called projection maps. For those maps the image has a Jordan algebra structure, which we consider in some detail. Special emphasis will be on projections onto the Jordan algebras called spin factors, showing in particular that those projection maps have quite special properties.
Erling Størmer

### Chapter 3. Extremal Positive Maps

Abstract
The positive maps of norm less than or equal to 1 form a convex set. It is natural to expect that the extreme points of this set have special properties. In Chap. 3 we study some of the main extremal maps, in particular Jordan homomorphisms and their applications to maps with strong extremal properties. In the last two sections we consider the relationship of the Stinespring theorem to extremal maps; first to the so-called non-extendible maps and then to a Radon-Nikodym theorem for completely positive maps together with the analogue of the GNS-representation for pure states.
Erling Størmer

### Chapter 4. Choi Matrices and Dual Functionals

Abstract
From linear algebra it is well known that linear maps between full matrix algebras are represented as matrices in the tensor product of the algebras, and also by linear functionals on the tensor product. In the study of positive maps the matrix associated with a map is called its Choi matrix and the associated functional its dual functional. We show how positivity properties of maps are reflected in positivity properties of their Choi matrices. For example, complete positivity of a map is the same as saying that its Choi matrix is positive, and similarly for the dual functional. We thus have a very useful technique for studying positive maps.
Erling Størmer

### Chapter 5. Mapping Cones

Abstract
In order to group together maps with similar properties we introduce certain convex cones of positive maps of the bounded operators B(H) on a Hilbert space into itself, called mapping cones. Then we define positivity of a map of a C -algebra into B(H) with respect to a mapping cone and show the basic properties of this positivity concept. For example we see that positivity for a map with respect to the smallest mapping cone is equivalent to its dual functional being a positive multiple of a separable state. We conclude by showing a Hahn-Banach type extension theorem for maps positive with respect to a mapping cone.
Erling Størmer

### Chapter 6. Dual Cones

Abstract
A cone in a Hilbert space has a natural dual cone consisting of vectors with positive inner product with vectors in the given cone. In the finite dimensional case we can do the same for cones of positive maps with respect to the Hilbert-Schmidt structure. Characterizations of maps in dual cones are given both in terms of their Choi matrices and their dual functionals. Furthermore, examples of dual cones of well known mapping cones like for example k-positive maps, are shown to be well known mapping cones. The ideas of dual cones are used in Sect. 6.3 to classify positive maps of the 2×2 matrices into themselves, and in the last section we characterize maps in dual cones in terms of their positivity properties on tensor products with maps in the given cone.
Erling Størmer

### Chapter 7. States and Positive Maps

Abstract
In this chapter we apply the results on dual functionals of positive maps to study states on tensor products of full matrix algebras. In particular we obtain characterizations of separable and PPT-states, and we also see how entanglement is related to the negative part of the Choi matrix of a positive map. Some of these ideas will be used to study the celebrated Choi map of the 3×3 matrices into themselves. Finally it is shown that the dual functional of the sum of the trace and a positive map of norm 1 is separable.
Erling Størmer

### Chapter 8. Norms of Positive Maps

Abstract
There are several different norms that can be introduced to positive maps. In this chapter we shall study some, which are closely related to mapping cones, and we show how positivity properties are reflected in norm properties of the maps. For the simplest example of completely positive maps, those of the form aV aV, we relate some of the norms of the map to the Ky Fan norms of VV .
Erling Størmer

### Backmatter

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