# 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.