Lecture Notes on Operator Algebras John M. Erdman Portland State University Version March 12, 2011 c 2010 John M. Erdman E-mail address:
[email protected] Contents 1 Chapter 1. LINEAR ALGEBRA AND THE SPECTRAL THEOREM3 1.1. Vector Spaces and the Decomposition of Diagonalizable Operators3 1.2. Normal Operators on an Inner Product Space6 Chapter 2. THE ALGEBRA OF HILBERT SPACE OPERATORS 13 2.1. Hilbert Space Geometry 13 2.2. Operators on Hilbert Spaces 16 2.3. Algebras 19 2.4. Spectrum 21 Chapter 3. BANACH ALGEBRAS 23 3.1. Definition and Elementary Properties 23 3.2. Maximal Ideal Space 26 3.3. Characters 27 3.4. The Gelfand Topology 28 3.5. The Gelfand Transform 31 3.6. The Fourier Transform 32 Chapter 4. INTERLUDE: THE LANGUAGE OF CATEGORIES 35 4.1. Objects and Morphisms 35 4.2. Functors 36 4.3. Natural Transformations 38 4.4. Universal Morphisms 39 Chapter 5. C∗-ALGEBRAS 43 5.1. Adjoints of Hilbert Space Operators 43 5.2. Algebras with Involution 44 5.3. C∗-Algebras 46 5.4. The Gelfand-Naimark Theorem|Version I 47 Chapter 6. SURVIVAL WITHOUT IDENTITY 49 6.1. Unitization of Banach Algebras 49 6.2. Exact Sequences and Extensions 51 6.3. Unitization of C∗-algebras 53 6.4. Quasi-inverses 55 6.5. Positive Elements in C∗-algebras 57 6.6. Approximate Identities 59 Chapter 7. SOME IMPORTANT CLASSES OF HILBERT SPACE OPERATORS 61 7.1. Orthonormal Bases in Hilbert Spaces 61 7.2. Projections and Partial Isometries 63 7.3. Finite Rank Operators 65 7.4.