This book tours the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, along with some of their main applications. It requires only a basic background in functional analysis. The presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will appreciate how the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature. An indispensible introduction to the theory of operator spaces.

1. Introduction; 2. Positive maps; 3. Completely positive maps; 4. Dilation theorems; 5. Commuting contractions; 6. Completely positive maps into Mn; 7. Arveson's extension theorems; 8. Completely bounded maps; 9. Completely bounded homomorphisms; 10. Polynomially bounded operators; 11. Applications to K-spectral sets; 12. Tensor products and joint spectral sets; 13. Operator systems and operator spaces; 14. An operator space bestiary; 15. Injective envelopes; 16. Multipliers and operator algebras; 17. Completely bounded multilinear maps; 18. Applications of operator algebras; 19. Similarity and factorization degree.