MATH 546 C1 —  Hilbert Spaces 

Time:  MWF 12:00-12:50 pm

Location:  DKH 206

Instructor:Marius Junge    Office hours:  tba

Course description:  Hilbert space theory is actually linear algebra in infinite dimension. As such applications of this theory can be found in geometry, ODE and PDE, operator algebras, and quantum information theory. Some basic understanding in functional calculus can make the difference between elementary and more sophisticated methods. 

We will not have a grader. My preferred methods is submission in pairs if we have to many students-lets discuss this in class. I also prefer presentations over finals. 

Part I: Basic Hilbert spaces theory
      1) Banach space and linear maps
      2) sesquilinear forms and Cauchy Schwarz,
Gram Schmidt, ONB and best approximation
Riesz representation
geometric properties
      6) Compact and Fredholm operators( Homework)


Part II: Basic Banach and C*-algebra theory and spectral
     1) Banach algebras with and without unit
      2) Spectrum of commutative Banach algebras
      3) Spectrum of commutative C*-algebras
      4) Application: Spectrum of compact normal operators
      5) Positive elements and positive functional
      6) approximate units and quotients
      7) Applications

Part III: Spectral theory for normal operators
    1) Motivation: Volterra operator, Shift operator
      2) Riesz Representation  theorem
      3) Projection valued measures 
      4) Spectral decomposition for normal operators
      5) Multiplicity theory

Part V: Hilbert C*-modules and completely positive maps
     1) Definition of Hilbert C*-modules
     2) Projective modules, W*-modules
     3) Completely positive maps and the  GNS representation
     4)  Kasparov’s representation  theorem
Further applications


John B. Conway:  A course in Functional Analysis, Springer, Graduate Text in Mathematics 1990 (second edition)

John B. Conway: A Course in Operator Theory, Graduate Studies in Mathematics, Vol. 21, 1999

E. Lance: Hilbert C*-modules,   

London Mathematical Society Lecture Note Series, 210., Cambridge University Press, Cambridge, 1995


Homework   (50%)
Final exam or  presentation  (50%)

Comments: I expect everybody to give a presentation, please look at the books and report on your topic before spring brake