Positive Linear Maps of Operator Algebras

1. Front Matter

   Pages I-VIII

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2. Generalities for Positive Maps

   • Erling Størmer

   Pages 1-12

3. Jordan Algebras and Projection Maps

   • Erling Størmer

   Pages 13-26

4. Extremal Positive Maps

   • Erling Størmer

   Pages 27-47

5. Choi Matrices and Dual Functionals

   • Erling Størmer

   Pages 49-62

6. Mapping Cones

   • Erling Størmer

   Pages 63-73

7. Dual Cones

   • Erling Størmer

   Pages 75-93

8. States and Positive Maps

   • Erling Størmer

   Pages 95-111

9. Norms of Positive Maps

   • Erling Størmer

   Pages 113-119

10. Back Matter

    Pages 121-134

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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 The author, a leading authority on the subject, sets out numerous results previously unpublished in book form.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.

• Choi matrices
• Jordan Algebras
• Positive maps
• completely positive maps
• mapping cones
• matrix theory

From the reviews:

“This book is a timely and valuable contribution to the theory and applications of positive linear maps defined on operator algebras and operator systems. … The style of writing is succinct and to the point. … the book’s value is in its content, the vast majority of which cannot be found in other monographs, and it will surely become the essential reference on positive linear maps for mathematicians and physicists alike.” (Douglas R. Farenick, Mathematical Reviews, January, 2014)

“The author, who is a reputed researcher in this discipline, gathers in this book the main results on positive operators between C*-algebras with the additional novelty of inserting results never before collected together in a book. … This book is … a very interesting contribution to the theory of positive linear maps from its beginnings to the latest results. It makes for interesting reading both from a mathematical point of view, as well as for those who are interested in quantum information theory.” (Antonio M. Peralta, zbMATH, Vol. 1269, 2013)

• Department of Mathematics, University of Oslo, Oslo, Norway

  Erling Størmer

Stormer's area of work is operator algebras. His main specialties have been non-commutative ergodic theory and positive maps. In connection with the latter the author has also worked on Jordan algebras of self-adjoint operators. He has received the main prize from the Norwegian Science Foundation, the Möbius Prize.

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