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. The spectral theorem for operators on Hilbert space and
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) G. Pedersen, Analysis Now, Springer Verlag, GTM 118, 1988,
2nd edition.
Further references to other textbooks and research articles will be given
throughout the course.
Syllabus:
The main objects of this course are C*-algebras and von Neumann
algebras, which are certain algebras of bounded operators on a Hilbert space.
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 already
had many deep applications to theoretical physics and geometry.
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. Then I plan to discuss many concrete examples such as
the non-commutative tori, which have come up recently in the context of
string theory. General ideas and operator algebras concepts will be
explained via these concrete examples.
The prerequisite for this course is the material covered in a first year
real analysis course at the graduate level. In particular
a basic knowledge of the elementary principles of functional analysis
will be assumed.
Grading: There will be no exams. The course grade will be based on
attendance, homework problems and a short presentation that each enrolled
student will give during the lectures. Topics for these presentations
will be distributed at the beginning of the course.