Fall 2015 MATH 206 001 LEC

Banach Algebras and Spectral Theory
001 LECMWF 11-12P 3 EVANSRIEFFEL, M A54460
Units/CreditFinal Exam GroupEnrollment
4NONELimit:25 Enrolled:19 Waitlist:0 Avail Seats:6 [on 10/04/15]
Additional Information: 

Prerequisites: 202A-202B or equivalent is more than enough. Students who have studied only part of the material of Math 202AB and wish to enroll in Math 206 should discuss this with me.

Syllabus: Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Analytic functional calculus. Hilbert space operators. C*-algebras of operators. Commutative C*-algebras. Spectral theorem for bounded self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. Hilbert-Schmidt operators. Fredholm operators. The Fredholm index. Selected additional topics.

Office: 811 Evans

Office Hours: TBA

Required Text: 

Recommended Reading: A Course in Functional Analysis, John B. Conway, 2nd ed., Springer-Verlag.

My understanding is that through an agreement between UC and Springer, chapters of the first edition of this book are available for free download by students. They should be satisfactory - I do not plan to assign exercises from the Conway book.

Course description: 

The theory of Banach algebras is a very elegant blend of algebra and topology which provides unifying principles for a number of different parts of mathematics and its applications, notably operator theory, commutative and non-commutative harmonic analysis and the theory of group representations, and the theory of functions of one and several complex variables. But at the present time probably its most extensive use is as a foundation for non-commutative measure theory (von Neumann algebras, Math 209) and non-commutative topology and geometry (C*-algebras, Math 208). These in turn provide a foundation for quantum physics, but they also have myriad applications in many other directions, including group representations and harmonic analysis, ordinary topology and geometry, and even number theory. (For a vast panorama of the applications see Alain Connes' 1994 book "Noncommutative Geometry", which is available on the web as a free download.) I will cover the standard topics as listed in the catalog. Beyond the basic general theory of Banach algebras this will include several forms of the spectral theorem for self-adjoint operators on Hilbert space, compact and Fredholm operators, and group algebras and the Fourier transform and their relation to representation theory. Further topics as time permits, probably including Toeplitz operators and their index. The problem sets will involve a number of important specific examples beyond those presented in class.

Homework: Many weeks I will give out a problem set, and the course grade will be based on the work done on these. There will be no final examination.

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