An Introduction to Operator Algebras

Description

Focuses on the fundamental results in operator algebras. This work discusses results including 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 and the functional calculus for normal operators. Content: 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[infinity] as a von Neumann Algebra -- 22. Abelian von Neumann Algebras -- 23. The L[infinity]-Functional Calculus -- 24. Equivalence of Projections -- 25. A Partial Ordering -- 26. Type Decomposition.