MATH 656 - Spring 2016
Project Topics
The purpose of this project is to read a short paper or sections from a book on a topic related to
the course. It should not be something that you learned in another course. I want a short write-up
of roughly 6 to 10 pages including references to all sources used. In addition, there will be a 50
minute presentation. Since this is a limited amount of time, you will have to summarize the topic.
Concentrate on definitions, examples, statements of results, concepts and key ideas rather than
attempting to give a complete proof. The paper should include these things plus the proofs in your
own words – do not just copy – fill in details to clarify the arguments. There is no need to do
everything in the paper or book section.
The list below contains possible topics along with some suggested references, but if you have
other ideas for a project, please do talk with me about it. They are not in any particular order.
Once you have chosen a topic. Let me know because only one person can do a particular item –
and it is first come, first served.
1. An introduction to K-Theory for C∗ -algebras. (See Chapter 7.1 in Murphy.)
2. Tensor products of C∗ -algebras: the maximal and spatial C∗ -tensor norm, minimality of the
spatial norm. (See Chapter 6.3-6.4 in Murphy.)
3. The algebra S1 (H) of trace class operators on a Hilbert space H: Show that S1 (H) is the
dual of the compact operators K(H) and B(H) is the dual of S1 (H). (See J. Conway’s A
course in operator theory and Chapter 2.4 in Murphy.)
4. Introduction to Toeplitz operators. (See Ken Davidson’s C∗ -algebras by Example, section V.1,
or Chapter 3.5 in Murphy.)
5. Crossed products of C∗ -algebras by group actions: the full crossed product, reduced crossed
product, integer actions. (See N. Brown and N. Ozawa’s C∗ -algebras and finite dimensional
approximations, Chapter 4.1-4.2).
6. Cohen’s factorization theorem: if A is a Banach algebra with a bounded approximate identity,
and X is an essential Banach A-module, then X = AX = {a·x : a ∈ A, x ∈ X}. (See Bonsall
and Duncan’s Complete normed algebras, Section 1.11.)
7. Amenable groups and Dixmiers Theorem: every bounded representation of a discrete amenable
group is similar to a unitary representation. (See V. Paulsen’s Completely bounded maps and
operator algebras, Theorem 9.3 and Ken Davidson’s C∗ -algebras by Example, Section V.II.2.)
8. Unbounded symmetric operators and the Cayley transform. (See G. Pedersen’s Analysis
Now, or Conway’s A course in functional analysis, or many books on applied functional
analysis.)
9. Abelian von Neumann algebras: Every abelian von Neumann algebra acting on a separable
Hilbert space can be represented as L∞ (Ω, µ), for some probability measure µ on a secondcountable compact Hausdorff space Ω. (See Chapter 4.4 in Murphy.)
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