Functional Analysis 2
(Summer term 2020)
News
- The lecture has been moved to 14:15-15:45 on Tuesdays.
- All material, including course notes, assignments and assignment solutions
will be distributed using
Moodle.
- Starting May 5, the course will be held online. If you would like to participate, send me an email with your name, your student ID and the semester when you took Functional Analysis 1.
Time and Place
Lecture:
Tuesday, 14-16, online
Thursday, 14-16, online
Lectures will be held using Zoom, for which participants have received an invitation by email.
The invitation can also be found on Moodle.
Exercise session:
To be determined.
Office Hour: Thursday, after class, or by appointment
Moodle
All material, including course notes, assignments and assignment solutions will be distributed using
Moodle.
To self-enrol for this course in Moodle, you need the enrolment key, which will be provided after
registration by email. You are encouraged to use
the discussion boards in Moodle for questions and discussion, both about general organization
and about mathematics.
Contents
This course will be a continuation of Functional Analysis 1, taught by Prof. Groves during the winter term 2019/2020.
Functional Analysis 2 will be divided into two parts, local convexity and operator algebras.
Locally convex spaces allow the study of notions of convergence that are not given by a norm, such as pointwise
convergence of functions. They are also a convenient framework for the discussion of the important notions
of weak and weak-* convergence.
The second part is concerend with the study of algebras of operators on Hilbert space.
On the one hand, studying such algebras leads to insights into the structure of single operators.
On the other hand, operator algebras are a fascinating topic in their own right, which
has implications to group theory and mathematical physics.
List of topics:
Part I:
- Locally convex spaces,
- weak topologies,
- the Hahn--Banach theorem for locally convex spaces,
- compactness and Alaoglu's theorem,
- extreme points and the Krein-Milman theorem.
Part II:
- Banach algebras and Gelfand theory,
- commutative C*-algebras and the Gelfand-Naimark theorem,
- states and the GNS construction,
- von Neumann algebras.
Prerequisites are Functional Analysis 1.
Course Announcement
Exam
There will be weekly assignments. You are expected to achieve at least half of the available points on the assignments.
At the end of the term, there will be an oral exam.
Literature
Part I:
Conway, John B., A course in functional analysis, 1990.
Lax, Peter D., Functional analysis, 2002.
Pedersen, Gert K., Analysis now, 1989.
Rudin, Walter, Functional analysis, 1991.
Werner, Dirk, Funktionalanalysis, 2000.
Part II:
Blackadar, Bruce, Operator algebras, 2006.
Davidson, Kenneth R., C*-algebras by example, 1996.
Dixmier, Jacques, C*-algebras, 1977.
Murphy, Gerard J., C*-algebras and operator theory, 1990.
Kadison, Richard V. and Ringrose, John R., Fundamentals of the theory of operator algebras. Vol. I, 1983.
Last update: May 07, 2020 Michael Hartz