Prerequisites:
Linear algebra. Basic real and functional analysis, i.e. the part of
analysis which is usually covered in a first year graduate course in
analysis and a second year course in functional analysis. The spectral
theorem for operators on Hilbert space, continuous and Borel functional
calculus (will be reviewed in the first week).
Recommended Literature:
There will be no textbook. The following books are recommended reading:
1) Kehe Zhu, An Introduction to Operator Algebras, CRC Press, 1993.
2) K. Davidson, C*-algebras by example, American Math. Soc., Fields
Institute Monograph No. 6, 1996.
3) V. Jones, V. Sunder, Introduction to Subfactors,
Cambridge University Press, 1997.
4) Masamichi Takesaki, Theory of Operator Algebras I, II, III,
Springer-Verlag 2002.
5) David Evans, Y. Kawahigashi, Quantum Symmetries on Operator
Algebras, Oxford University Press, 1998.
6) Serban Stratila, Laszlo Zsido, Lectures on Von Neumann Algebras.
Additional references will be given throughout the course. Several
lecture notes on operator algebras from various authors are available
online, e.g. Vaughan Jones' course on
von Neumann algebras, see https://math.berkeley.edu/~vfr/.
Syllabus:
The main objects of this course are C*-algebras and von Neumann
algebras.
Every abelian C*-algebra with unit is an algebra of continuous functions
on a compact Hausdorff space and two such algebras are isomorphic
iff the underlying spaces are homeomorphic. Noncommutative C*-algebras
can therefore be viewed as algebras of ``functions'' on
``noncommutative spaces'' and noncommutative algebraic topology
(called K-theory or KK-theory) can be developed to study these spaces.
Von Neumann algebras are noncommutative measure spaces and ideas from
ergodic theory and probability theory play an important role in studying
these objects. The idea of replacing a space by a ``quantum space'',
i.e. a naturally associated operator algebra, is at the basis of Alain
Connes' noncommutative geometry, a mathematical theory, which has
many applications to mathematics and physics.
This course is an introduction to the theory of operator algebras, and
I will start with a discussion of the basic properties and constructions
of a C*-algebra and a von Neumann algebra. This will include positivity,
states, the GNS construction, von Neumann's bicommutant theorem, Kaplansky's
density theorem, comparison theory of projections in von Neumann
algebras, von Neumann type of factors etc.
I plan to discuss many concrete examples, such as the CAR algebra,
the Cuntz algebras, non-commutative tori and group von Neumann algebras.
General ideas and operator algebras concepts will be explained via these
concrete examples.
Grading: There will be no exams. The course grade will be based on
attendance. I will give exercises during the lectures, and there may be
team presentations
on a particular operator algebra topic of your choice. I will assign the
teams and distribute a list of possible topics, but you are welcome to
propose your own topic.