MRI Course on C*-Algebras, Autumn 2009

MRI Course on C*-Algebras, Autumn 2009


The oral exams will take place from January 26 (Tuesday) to 29 (Friday) in Utrecht. They will last between 30 and 45 minutes each and will start at the times 11 am, 1, 2, 3, 4 and (if necessary) 5 pm.

Everyone who would like to take the exam is invited to contact me a.s.a.p. and tell me which of the free time slots in the table below (s)he would like. I will then insert the names in a first-come-first-served manner. (Should no suitable slot be left, I will try to find a possible substitution.)


TimeJan 26Jan 27
Jan 28Jan 29
11 amTessa van der HoornRalph LangendamKaYin LeungJason Moss
1 pmRob EggermontJavier SaenzRoy WangMarcin Lis
2 pmGijs HeutsBoris Osorno TorresMaciej Koch-JanuszAlex Zamudio
3 pm---Kasimir van RijnJuan David Barrera CanoOri Yudilevich
4 pm---Chris van DorpEric SieroMichael Agathos
5 pm------Daniel van Dijk---

Students that have not yet submitted the EVALUATION FORMS can find them here
and send them to the following address:
Secretary Regieorgaan Masteropleidingen Wiskunde, Mrs. Greta Oliemeulen-Löw
Radboud University Nijmegen, FNWI, Heyendaalseweg 135, 6525 AJ NIJMEGEN

Wednesday 14:00-17:00, BBL Building room 276 (Buijs Ballot Lab)

Literature: We will mainly use the following book:

[D]: Kenneth R. Davidson: C*-Algebras by Example.
American Mathematical Society 1996.

Further recommended references:

[R]: Volker Runde: A taste of topology. Springer Universitext, 2005, 2008.
[P]: Gert Pederden: Analysis NOW. Springer Graduate Texts in Mathematics 118, 1989, 1996.
[M]: Gerald Murphy: C*-Algebras and operator theory. Academic Press, 1990.
[T]: Masamichi Takesaki: Theory of operator algebras I. Springer Encyclopedia of Mathematics, vol 124, 1979, 2002.
[L]: N. P. Landsman: Lecture notes on C^*-algebras.

Lectures begin on Wednesday, September 09. The last lecture will be given on December 16. No lecture on November 4.

The exams will take place as oral exams during the last week of January 2010.

Subjects treated (with references to books if different from Davidson):

09.09.: Introduction.
16.09.: Spectrum, Gelfand transform. Homework I.4, I.7.
23.09.: Gelfand transform for commutative C*-algebras. Positivity. Homework: I.1, I.8.
30.09.: Categories and functors; categorical formulation of Gelfand duality [L, Sect. 6]. Non-compact case: Proper maps. Some comments on holomorphic functional calculus [T, p.9-12]. Continuous functional calculus and spectral mapping theorem [D,M]. Square roots of positive elements. Homework: Make sure you understand categorical Gelfand duality. And: I.14.
07.10.: Positivity, approx. units. (I followed section I.4 of Davidson quite closely, except that I put more emphasis on operator monotone functions, e.g. in the proof of directedness of A_+,1.) Homework: I.5, I.9, I.10.
14.10.: Ideals in C*-algebras, quotient algebras, automatic continuity/isometry. States.
21.10.: More on states. Representations of C*-algebras (non-degenerate, cyclic, irreducible).
28.10.: GNS representation and GNS vector (`vacuum'). Universal representation. GNS rep. is irred.<>state is pure. Irreps separate points. Pure states on abelian C*-algebras are precisely the characters. Basics of compact operators. Exercises: I.26, I.33, I.36, I.39.
04.11.: NO LECTURE.
11.11.: Representations of ideals. More on compact operators and representations of algebras of compact operators.
18.11.: End of representations of compact operators. Finite dim. C*-algebras, AF-algebras. Inductive limit of C*-algebras. UHF-algebras.
25.11.: AF-algebras, using short text by Bratteli on AF-algebras and Bratteli diagrams.
02.12.: Ideals in AF-algebas. Classification of UHF-algebras. Basics on non-commutative torus.
09.12.: The Toeplitz algebra (algebra generated by the one-sided shift) and its universal property.
16.12.: Cuntz algebra. Definition of graph C*-algebras according to some chapters on graph C*-algebras.