Mastermath Couse OPERATOR ALGEBRAS (Fall 2015/6)

BRANDNEW: Mastermath has informed me of the rule that only integers are accepted as marks. I have changed the results accordingly.
This has led to both higher and lower marks (but no change to passing or not since 5.5 had been excluded before).
My apologies: I find this rule very bad.

Results of the exam of January 13, 2016 and final marks

Exam of January 13, 2016 with solution and comments.

Lectures: Michael Müger (Radboud Univ. Nijmegen) www
e-mail: mueger ## math.ru.nl

Exercise class: Sohail Sheikh (Radboud Univ. Nijmegen)
e-mail: S.Sheikh ## math.ru.nl

There will be homework every second week. The homework will determine (at least) 50% of the final mark, thus do it! Final examination will most likely be take home, i.e. a longer set of problems, but I reserve the right to hold oral exams.

The course is mainly based on this book: Gerard J. Murphy: C*-Algebras and operator theory, Academic Press 1990,
but occasionally I will use also other sources. In this cases, the material will be provided on this website.

The lectures take place in the Freudenthal building (Math Institute), room 611AB.

Subjects treated

LECT. 01 (16.09.): Generalities, prerequisites, some history and motivation. Then pages 1-9 in Murphy.
LECT. 02 (23.09.): Murphy p.9-16.
HOMEWORK set 1 (Based on Exercises 1, 5(a+b), 6 and 8 in Chapter 1 of Murphy.)
LECT. 03 (30.09.): Murphy p.16-18 (with more detail on the Wiener/Gelfand theorem, and an example of a Banach algebra each element of which is quasi-nilpotent). Then p.35-40. I skipped the multiplier algebra (p.38-39) for now and gave a more direct proof of Theorem 2.1.6, following Proposition I.1.3 here.
LECT. 04 (07.10.): I covered pp. 41-46, up to and including Lemma 2.2.3 (occasionally with more detail than given in Murphy).
HOMEWORK set 2 (deadline: 21.10.)
LECT. 05 (14.10.): Rest of Section 2.2. Then Section 3.1 up to Thm. 3.1.3.
LECT. 06 (21.10.): Theorems 3.1.4-3.1.7, then Section 3.3 up to and including Theorem 3.3.6.
HOMEWORK set 3a (deadline: 04.11.) NOTE: This set is shorter than usual and will be continued next week!
LECT. 07 (28.10.): Rest of Section 3.3 (plus Thm. 2.3.5). Comments on Jordan decomposition of self-adjoint functionals (without proof, cf. Thm. 3.3.10). Section 3.4. Non-degenerate actions (Rem. 4.1.4). Then from Sect. 5.1: Thm. 5.1.1 and 5.1.3.
HOMEWORK set 3b (deadline: 11.11.)
LECT. 08 (04.11.): Murphy 2.3.5, 5.1.2, 5.1.4-5.1.8.
LECT. 09 (11.11.): Coro. 5.1.9 - Thm. 5.1.13. Then Sect. 4.1 up to Thm. 4.1.5.
HOMEWORK set 4 (deadline: 25.11.)
LECT. 10 (18.11.): Examples 2.5.1 and 4.1.2, Lemma 4.1.6-Thm. 4.1.10. Discussion of polar decompos. and Borel functional calculus.
NO EXERCISES THIS WEEK, BUT LOOK AT SECTION 2.5!! (Exam. 2.5.1 was done in the lecture.)
LECT. 11 (25.11.): End of Section 4.1. Hilbert-Schmidt and trace class operators (Murphy section 2.4 or Pedersen).
HOMEWORK set 5 (deadline: 09.12.)
LECT. 12 (02.12.): Rest of Section 2.4. Then Section 4.2 up to 4.2.8.
LECT. 13 (09.12.): Rest of Section 4.2. Then Sections 4.3 and 5.2 (except proof of Thm 5.2.2).
HOMEWORK set 6 A mistake in 3b has been fixed! (deadline: 04.01.)
LECT. 14 (15.12.): Proof of Thm 5.2.2. Classification of fin.-dim. C*-algebras (5.6.2, 6.2.1, 6.3.8). The multiplier algebra (more than in Murphy).

!!! EXAM !!! There will be a written exam on January 13. You will be allowed to use Murphy's book.