MTH 831 (Spring 2007): C*-algebras

Instructor: Hanfeng Li

Office: 104 Mathematics Building.     Phone: 645-6284 ext. 126.

Office Hours: F 3-4pm

Lectures: MWF 2:00-2:50pm

Room: 235 Math
Course Description
The theory of operator algebras (including C*-algebras and von Neumann algebras) grew out of the development of quntum mechnics, but now has interaction with many areas of mathematics.  We shall develop first the basic theory of C*-algebras (spectrum, functional calculus, positive elements, approximate units, states, GNS construction, ideals, homomorphisms, etc.), and then discuss briefly "noncommutative algebraic topology" (K-theory).  If time allowed, we shall also talk about "noncommutative differential geometry" (cyclic homology and spectral triples).

I shall discuss some important examples, including AF algebras, Cuntz algebras (more generally, Cuntz-Krieger algebras), and noncommutative tori (more generally, twisted group C*-algberas), coming from mathematical physics, dynamics, and group representations.
Recommended Reading
         The standard references for C*-algebra theory include:
           C*-algebras     by Dixmier,
           C*-algebras and W*-algebras     by Sakai,
           An invitation to C*-algebras     by Arveson,
           C*-algebras and their Automorphism Groups    by Pedersen,
           C*-algebras and Operator Theory    by Murphy,
           Fundamentals of the Theory of Operator Algebras    by Kadison and Ringrose,
           C*-algebras by Example    by Davidson,
           Theory of Operator Algebras    by Takesaki.
         I shall not follow closely any of these books. However, the choices of our topics will be close to that of Davidson's book.
Prerequisite
Some functional analysis, including Hilbert spaces, Banach spaces, bounded linear operators on Hilbert spaces, dual spaces, weak-* topology, Hahn-Banach theorem.  Some knowledge about algebraic topology or differential geometry will be helpful, but I shall not assume that.
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