An Introduction to Operator Algebras

An Introduction to Operator Algebras is a concise text/reference that focuses on the fundamental results in operator algebras. Results discussed include Gelfand's representation of commutative C*-algebras, the GNS construction, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) functional calculus for normal operators, and type decomposition for von Neumann algebras. Exercises are provided after each chapter. Preface, I. Banach Algebras, 1. Review on Functional Analysis, 2. Banach Algebras and the Invertible Group, 3. The Spectrum, 4. Multiplicative Linear Functionals, 5. The Gelfand Transform and Applications, 6. Examples of Maximal Ideal Spaces, 7. Non-Unital Banach Algebras, II. C*-Algebras, 8. C*-Algebras, 9. Commutative C*-Algebras, 10. The Spectral Theorem and Applications, 11. Further Applications, 12. Polar Decomposition, 13. Positive Linear Functionals and States, 14. The GNS Construction, 15. Non-Unital C*-Algebras, III. Von Neumann Algebras, 16. Strong- and Weak-Operator Topologies, 17. Existence of Projections, 18. The Double Commutant Theorem, 19. The Kaplansky Density Theorem, 20. The Borel Functional Calculus, 21. L°° as a von Neumann Algebra, 22. Abelian von Neumann Algebras, 23. The -Functional Calculus, 24. Equivalence of Projections, 25. A Partial Ordering, 26. Type Decomposition, Bibliography, Index