| Instructor | | Volker Runde |
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| Office hours | | N/A |
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| Course Content | |
The theory of operator algebras, i.e. of closed, self-adjoint algebras
of bounded linear operators on Hilbert space, has its origins in the
1930s with the work
of Gelfand, Neumark, von Neumann, and others. Even decades after its
inception, it continues to be an extremely active area of research with
numerous
connections to other branches of mathematics such as ring theory,
algebraic topology, differential geometry, and even computing (quantum
computers). |
| | This course is an introduction to the theory of operator algebras. I will certainly cover the following topics:
- Banach algebras: Definition and elementary properties of the
spectrum; spectral radius formula; Gelfand theory; the commutative
Gelfand-Naimark theorem.
- Operator theory: spectral theorem for normal bounded operators; Fredholm operators.
- C*- and von Neumann algebras: positivity; ideals and quotients; the GNS-construction; von Neumann's bicommutant theorem; Kaplansky's density theorem.
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| | Further topics will be covered depending on
the participants' background and the time avaialalbe. Possibilities are:
holomorphic functional calculus; bounded approximate identities; type
classification of von Neumann algebras; applications to harmonic
analsysis. |
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| Prerequisites | | MATH
516 (Linear Analysis) or equivalent and some familiarity with algebraic
concepts such as algebras, ideals, etc., are sufficient as
prerequisites for this course.
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| Textbooks | | There is no required text, but the following books are useful supplementary reading:
- J. B. Conway, A Course in Functional Analysis. Springer Verlag, 1985.
- J. B. Conway, A Course in Operator Theory. American Mathematical Society, 2000.
- K. R. Davidson, C*-Algebras by Example. American Mathematical Society, 1996.
- R. G. Douglas, Banach Algebra Techniques in Operator Theory. Springer Verlag, 1998.
- R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, I. American Mathematical Society, 1997.
- B.-R. Li, Introduction to Operator Algebras. World Scientific, 1992.
- G. J. Murphy, C*-Algebras and Operator Theory. Academic Press, 1990.
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| Grading | | The grade will be based on (approximately) biweekly homework assignments (50%) and a take home final (50%). |
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