Completely bounded maps and dilations.

*(English)*Zbl 0614.47006
Pitman Research Notes in Mathematics Series, 146. Harlow, Essex, England: Longman Scientific & Technical. Copubl. in the United States with John Wiley & Sons, Inc., New York. IX, 187 p. £15.00 (1986).

This book gives a self contained exposition of the theory of completely positive and completely bounded maps between \(C^*\)-algebras, and the application of this theory to similarity and dilation theory. A particular emphasis is given on applications to operator theory. Dilation theory is presented from an algebraic point of view.

Contents: 1. Introduction; 2. Positive Maps; 3. Completely Positive Maps; 4. Dilation Theorems; 5. Completely Positive maps into \(M_ n\); 6. Averson’s Extension Theorems; 7. Completely Bounded Maps; 8. Completely Bounded Homomorphisms; 9. Applications to K-spectral Sets; 10. Tensor Products and Joint Spectral Sets.

At the end of each chapter there are many exercises.

Contents: 1. Introduction; 2. Positive Maps; 3. Completely Positive Maps; 4. Dilation Theorems; 5. Completely Positive maps into \(M_ n\); 6. Averson’s Extension Theorems; 7. Completely Bounded Maps; 8. Completely Bounded Homomorphisms; 9. Applications to K-spectral Sets; 10. Tensor Products and Joint Spectral Sets.

At the end of each chapter there are many exercises.

Reviewer: I.Vidav

##### MSC:

47A20 | Dilations, extensions, compressions of linear operators |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

46L05 | General theory of \(C^*\)-algebras |