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C*-algebras

Vorlesung im Sommersemester 2006 an der TU Chemnitz (in englischer Sprache)
Lecture in the Summer term 2006

Coordinates:
Tba

C*-algebras are of interest as purely mathematical structures as well as for applications, e.g. in statistical physics and quantum physics. As G.K. Pedersen writes: "The theory of C*-algebras is the study of operators on Hilbert space with algebraic methods."

Content:
1. Basic notions and examples
2. Spectral theory and Gelfand transform
3. Commutative C*-algebras, functional calculus and
   automatic continuity
4. Normal operators in Hilbert spaces, the spectral theorem
  and spectral representation of commutative C*-algebras
5. Positivity and the GNS-construction
6. Von Neumann's double commutant theorem
7. The spectral theorem for unbounded selfadjoint operators
8. Projections in von Neumann algebras
9. Ordering of projections
10. Cyclicity, separation and countable decomposability
11. Finite and infinite projections
12. Type decomposition of von Neumann algebras
13. Group von Neumann algebras

Got questions, see ... here

References:

Kadison, Richard V.; Ringrose, John R.:
Fundamentals of the theory of operator algebras. Vol. I. Elementary theory.
Pure and Applied Mathematics, 100. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

Kadison, Richard V.; Ringrose, John R.:
Fundamentals of the theory of operator algebras. Vol. II. Advanced theory.
Pure and Applied Mathematics, 100. Academic Press, Inc., Orlando, FL, 1986.

Pedersen, Gert K.:
C*-algebras and their automorphism groups.
London Mathematical Society Monographs, 14. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.

S. Haeseler: Skript Hilberträume ps-file

Some old classics:

von Neumann, J.:
Über adjungierte Funktionaloperatoren. (German) Ann. of Math. (2) 33 (1932), no. 2, 294--310.

Murray, F. J.; Von Neumann, J.:
On rings of operators. Ann. of Math. (2) 37 (1936), no. 1, 116--229.

Murray, F. J.; von Neumann, J.:
On rings of operators. II. Trans. Amer. Math. Soc. 41 (1937), no. 2, 208--248.

v. Neumann, J. On rings of operators. III:
Ann. of Math. (2) 41, (1940). 94--161.

... to be continued ...