# Mastermath Couse OPERATOR ALGEBRAS (spring 2014)

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

The course is mainly based on this book: Gerard J. Murphy: C*-Algebras and operator theory,
but occasionally I will use also other sources.

## Subjects treated

05.02.: Banach algebras I (roughly pages 1-9 in Murphy). No exercises this week.
12.02.: Banach algebras II (pages 10-18 in Murphy).
HOMEWORK 1: exercises 1,5,6,10 in chapter 1 of Murphy.
19.02.: Some final comments on Banach algebras: L1(G) for G locally compact abelian. f(a) for f entire. Then roughly pp 35-40, plus a more direct discussion of the C*-norm on A~.
26.02.: Pages 41-45.
HOMEWORK 2: nos. 1,2,3,7,8 on p. 73-74
05.03.: Pages 46-47 and 77-80 up to Thm. 3.1.4. Plus: comments on operator monotone functions, construction of a commutative Banach algebra for which the Gelfand representation is identically zero, a direct proof of ||a||=r(a) for normal elements not using Gelfand transform, and proof of: If a is normal and the spectrum is contained in the reals (in the circle) then a is self-adjoint (unitary).
12.03.: Thms. 3.1.5-3.1.7, sect. 3.3 up to Thm. 3.3.8, sect. 3.4 up to Thm. 3.4.1.
HOMEWORK 3: exercises 1,2,3,4 in chapter 3 of Murphy. Plus these two.
19.03.: NO LECTURE, only exercise class (beginning at 2 pm)
26.03.: More on multiplier algebras (Murphy p.81 (after Remark 3.1.3) - p.83, including some reminders on K(H)). Self-adjoint linear functionals (p.92). Thm. 3.4.1-Thm. 3.4.3, incl. Thms. 2.3.5, 2.3.6. Definitions of non-degenerate repres. and cyclic vectors.
HOMEWORK 4: Exercise 8 on p.109, plus these three.
02.04.: Section 5.1 up to p.146.
09.04.: Section 5.1: 5.1.8 - 5.1.13, incl. discussion of locally convex spaces, convexity, faces, extreme points and the Krein-Milman theorem (cf. Appendix A) or here. Then Section 4.1 up to Thm. 4.1.5.
HOMEWORK 5: Exercises 4.1 and 4.2 in Murphy, plus these.
16.04.: NEITHER LECTURE NOR EXERCISES. BUT: Read up on trace-class and Hilbert-Schmidt operators, either Murphy from p.60 to the end of Section 2.4 or Pedersen,
23.04.: Sect. 2.5 (Borel functional calculus), Lemma 4.1.6 - Thm. 4.1.12.
30.04.: End of section 4.1 (plus some material on hereditary subalgebras and on ideals in B(H)). Section 4.2.
HOMEWORK 6: Exercises 4,5,6 on p.137-8 of Murphy, plus these two.
07.05.: Section 4.3 (Kaplanski density theorem), Section 5.2 (Transitivity theorem).
14.05.: Structure of finite dimensional C*-algebras. Section 6.1 and 6.2 up to Coro. 6.2.4.
HOMEWORK 7: Exercises 1 and 2 on p.213/214 plus these two.
21.05.: Last lecture: Rest of Section 6.2, plus more on AF-algebras: Bratteli diagrams. VNA of a discrete group. Theorem 4.3.4 (with a better proof), Section 4.4.