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 (including
the spectral theorem for normal operators on Hilbert space). Some knowledge
of basic operator algebras, continuous and Borel functional calculi would
be helpful, but is not required. I will review material if needed and/or
provide references.
Recommended Literature:
There will be no textbook. The following books are recommended reading:
1) John B. Conway, A Course in Functional Analysis,
Springer GTM 96, 2nd edition (January 1997).
2) John B. Conway, A Course in Operator Theory, Graduate
Studies in Mathematics, American Math. Soc. (1999).
3) Michael Reed, Barry Simon, Functional Analysis (Vol. I),
Academic Press, 1980.
4) Nate Brown, Narutaka Ozawa, C*-Algebras and
Finite-Dimensional Approximations, Graduate
Studies in Mathematics, Vol 88, American Math. Soc. (2008).
5) Vern Paulsen, Completely Bounded Maps and Operator Algebras,
Cambrdige Studies in Advanced Mathematics 78 (2002).
6) Kehe Zhu, An Introduction to Operator Algebras, CRC Press, 1993.
7) Serban Stratila, Laszlo Zsido, Lectures on Von Neumann Algebras,
Abacus Press 1979.
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:
This course will cover several topics from functional analysis that typically
are not standard material in Math 7120. The course will start with a
discussion of Hilbert-Schmidt and trace class operators, noncommutative
Lp-spaces and noncommutative duality. As an application,
we will see that every von Neumann algebra is a dual Banach space.
I may spend some time on basic Fredholm theory, if there is interest.
In the next part of the course, I plan to cover basics of unbounded
operators with possibly some applications to Tomita-Takesaki theory.
Then, the course will shift towards topics from quantum information
theory. This may include a discussion of quantum channels
(completely positive maps), the notion of entanglement and
connections to Connes' embedding problem, if time allows. The
role of commuting squares from subfactor theory in quantum
information may be discussed in this section as well.
In the next part of the course, I plan to discuss approximation and
rigidity properties of groups and operator algebras. This would include
various definitions of amenability of a group,
the Haagerup approximation property and property (T)
of Kazhdan. What exactly we end up discussing will depend on time and student
interest.
Grading: There will be no exams. I will assign problems on a regular
basis and, depending on the number of students enrolled, there will be
presentations by teams on a topic of your choice. I will select the
teams and distribute a list of possible topics, but you are welcome to
propose your own topic.
The course grade will be based on attendance and the team presentation.